The tourists steadily took top order wickets in the first two sessions before taking the final five after tea.
Moeen Ali claimed 4-71 and Jack Leach 3-60, while Adil Rashid and Ben Stokes picked up a wicket each.
The win puts England 1-0 up in the three-match series and is their first in an overseas Test since October 2016.
Joe Root savours first Test away win as captain after rout of Sri Lanka
Between them, Englands three spinners took 16 wickets in the match, the second best return by English spinners in a Test since 1958.
“Weve some difficult selections ahead, but Im not complaining about that. Its great to see the strength in depth in this squad. Im going to enjoy that difficult selection meeting. It is going to be tough. But thats what you want. You want guys performing, you want to make those tough calls.
LIVE SCORECARD: Follow the fourth day of the first Test between Sri Lanka and England in Galle
A dominant performance from England over the first three days had left Sri Lanka needing to bat out the final two days to draw – or score a record 462 to win.
Then again, Rashid made key contributions. A punchy 35 put the gloss on Englands work on the opening day. Then, he broke a 75-run partnership between Chandimal and Angelo Mathews in the first innings when the other bowlers were unable to make an incision. Some of the full tosses and long hops on day four were comical. At one point, he even bowled himself into the pitch. It is likely he will miss out for a seamer – perhaps Stuart Broad – when the series moves on to Kandy.
Beginning the day 15-0, the hosts survived the first hour unscathed but lost three quick wickets before lunch.
Angelo Mathews offered resistance with 53 but he chipped Moeen to mid-wicket shortly after tea as England closed in.
“It is right up there as one of my best wins,” said Root. “We were in control for the majority of the game which shows how skilful and consistent we have been.
“As a group we have all contributed in some form but to see young inexperienced guys put together really important match-wining performances is really pleasing. It shows the plans are right, the preparation is right.”
Once again the England spinners did most of the work and took most of the wickets though there was one excellent, hostile spell from Stokes either side of lunch. As in the first innings Moeen Ali ended up with the most victims – another four – and he finished with his best match figures outside of England (eight for 137). When he dismissed Akila Dananjaya he equalled the haul (153) of one of the great English off-spinners, Fred Titmus. In the off-spinning department there is only Jim Laker (193) and Graeme Swann (255) left for Moeen to pursue. As with John Emburey, whom he overtook in the first innings, Moeen has bowled 5,000 fewer deliveries than Titmus ever did, which reflects how the game has changed.
Englands bowling attack has struggled for potency overseas in recent years but in Galle, Root was able to call on a variety of bowlers who all contributed.
Root started day four with seamers James Anderson and Sam Curran before turning quickly to spin when there continued to be no swing.
Leaving aside the scale of Englands victory the match did not quite follow the anticipated pattern since runs were much easier to come by than is often the case here. The ball did not spin as venomously as in Sri Lankas last match in Galle against South Africa in July. Maybe the damp weather before and during this match was a factor. It was miraculous after torrential overnight rain that play could start on time but the system of covering every blade of grass on the playing area had worked superbly again. Maybe the groundstaff were sympathetic to the good traders of Galle who welcome the advent of so many overseas visitors eager to eat, drink and be merry.
He first introduced left-arm spinner Leach, followed by off-spinner Moeen, and the pair removed the Sri Lanka openers, with Leach trapping Kaushal Silva lbw for 30 and Moeen catching Dimuth Karunaratne off his own bowling for 26.
All-rounder Ben Stokes had a light workload with the ball in this Test but he dismissed Dhananjaya de Silva on the stroke of lunch and bowled a hostile spell after the interval.
He peppered Mathews and Kusal Mendis with excellent short bowling and should have had been rewarded with another wicket but Anderson dropped a simple catch from Mathews.
Leach bowled a long, accurate spell in the afternoon session and claimed the two wickets to fall, having Mendis caught at mid-off and bowling Sri Lanka captain Dinesh Chandimal with a brilliant spinning delivery.
Moeen then returned after tea and for the second time in the match took a wicket with the first ball of the final session, this time having Dhananjaya de Silva caught by Stokes at slip.
Foakes described his sudden elevation to Test cricket as an unbelievable place to be. He has not had time to let everything sink in. He was asked about Kandy. Jonny will be fit potentially. He kept thinking. So I guess hes the number one and if I play one game Im quite happy with that one. Somehow that is unlikely to be the case.
England beat Sri Lanka by 211 runs inside four days at first Test – as it happened
Shortly afterwards Moeen saw off Mathews and leg-spinner Rashid claimed his first wicket of the innings by dismissing Dilruwan Perera for an aggressive 30, before the victory was sealed by the run out of Herath.
Sri Lanka v England: Keaton Jennings scores first century since debut
Other than Root, who bowled one over on day four, every England bowler took a wicket in the match but it will be the spinners who pleased the England captain most.
Their combined 16 wickets is the second-best spinners haul for England for 60 years, beaten only by the 19 Graeme Swann and Monty Panesar shared in Mumbai in 2012.
The opening batsman was ably assisted by Ben Stokes, who scored a half-century, and handy thirties from Jos Buttler and Ben Foakes as England amassed 322 for six declared.
“They controlled when required, attacked when they could and I always felt we would take wickets throughout the game.”
Englands losing run overseas stretched back all the way to 30 October 2016 when they lost to Bangladesh in Dhaka.
Cricket: Michael Vaughan names his England team for second Sri Lanka Test
Since then England have lost series 4-0 in both India and Australia, plus 1-0 in New Zealand earlier this year, with three draws accompanying the nine defeats.
Previously at Galle England had drawn twice and lost twice but on this occasion, after being reduced to 103-5 before lunch on day one, they then dominated the match.
England demonstrate strength of spin attack in finally cracking the Galle citadel
As well as the bowlers the batsmen also contributed, with man-of-the-match Ben Foakes and Keaton Jennings hitting centuries in the first and second innings respectively.
“To be able to contribute to the team is just amazing,” Foakes said. “It has been one of the best weeks of my career so far, if not the best.
England secured a significant victory in their first Test against Sri Lanka, sealing a 211-run win after another barn-storming display from spin-bowler Moeen Ali.
Sri Lanka vs England: James Anderson handed demerit point for dissent in Galle test
A win in the second Test in Pallekele, starting on Wednesday, would seal Englands first away series win since beating South Africa in 2016.
Their bowlers were not as effective as Englands, their fielding disappointed and some of their batsmen again fell to reckless shots on the fourth day.
Making an albeit unlikely attempt to save the Test, Karunaratne was out attacking Moeen, hitting the ball straight back for a catch in the morning session, and Mendis foolishly charged down the wicket and skewed a shot to mid-off.
The ball before Dhananjaya edged to Stokes at first slip in the final over before lunch, he had played a wild drive to the Durham all-rounder which had been given out caught behind, only to be overturned on review.
Sri Lanka v England: Keaton Jennings played Alastair Cook-like innings
Heraths dismissal – sprawled in the dirt after failing to beat Stokes powerful throw – was an ignominious finish for a player who ended with 433 Test wickets – putting him joint eighth on the all-time list – but also a strange one as he had called for the risky second run.
England player ratings from first-Test victory against Sri Lanka – who has cemented his place for Kandy clash?
The 40-year-olds team-mates celebrated his retirement at the end of play by carrying the spinner around his home ground on their shoulders but Sri Lanka will now have to do without him for the remaining two Tests.
“Its a hard day for us, because we know how much Rangana has done for the team and Sri Lanka cricket,” reflected Sri Lanka captain Dinesh Chandimal, who struggled in this match with a groin injury sustained while fielding.
“Credit goes to England, they played some outstanding cricket. We had a really good start in the first session but we couldnt capitalise on it.”
England spinner Leach made Chandimal (1) his third victim of the day after pitching the ball around leg stump and then spinning it back to hit off.
Former England captain Michael Vaughan on The Cricket Social podcast: “Its been a tremendous performance.
Jennings gives England huge lead
“They have been testing conditions and after the first session on the first day, England produced a wonderfully professional performance.
LIVE SCORECARD: Follow the third day of the first Test between Sri Lanka and England
“They played proper Test cricket and it was the youngster Ben Foakes who led the way, then Keaton Jennings scored a hundred in the second innings.
England set Sri Lanka huge target after Jennings ton
“The variations Joe Root has got in the field are wonderful and England can be very proud of the way they played.
Katherine Brunt in doubt for Englands World T20 opener against Sri Lanka – ESPNcricinfo
“I hope they are honest, though – yes, celebrate a win but be realistic that a better team wouldnt have allowed them back into the Test match after that first morning.
“This England team, though, has got everything it needs to be an outstanding Test match team – they have power, variations, three spin options and they just need to find the right mentality at the top of the order and read situations better.
“I suppose they dont have a 95mph bowler, but not many sides do. I just hope they use this opportunity to build some momentum this winter, against Sri Lanka and West Indies, heading into next summers Ashes series.”