Leach took three of the six required wickets as well as a vitally important run out to seal a 42-run win and make Joe Root's side just the third English touring team to sweep a series of three or more games.
Such achievements are not easy to come by – the last time it happened, in New Zealand, John F Kennedy was planning for a second term as President of the United States and the Beatles were three days away from releasing their debut album – and Sri Lanka pushed England harder than anyone expected.
— England Cricket (@englandcricket) November 26, 2018
Chasing a ground record 327 and starting on 53 for four, Kusal Mendis (86) and Roshen Silva (65) both made major inroads before a last-wicket stand worth 58 threatened the unlikeliest of turnarounds in the third Test.
Leach's left-arm spin earned him figures of four for 72, and the match-winning wicket of Malinda Pushpakumara four balls after tea, but it was a right-arm throw that did the most damage. He rattled the stumps from 40 yards to see off Mendis and break an ominous 102-run partnership with Roshen.
Hopes of a quick clean-up job on day four were quashed early on as Mendis steered Sri Lanka through the morning session almost unscathed. By lunch they had scored 111 runs and lost just one wicket – nightwatchman Lakshan Sandakan lingering for half-an-hour before Leach got one to drift and clip the edge.
England's spinners had otherwise endured one of their worst sessions since arriving on the island, Moeen Ali and Adil Rashid regularly releasing the pressure with short balls or full tosses. Mendis pounced on any error in length with Roshen busily building his supporting role.
There were 10 boundaries in the morning's play as well as one early six, Mendis pumping a loosener from Moeen over mid-wicket. Root kept the vastly experienced Stuart Broad out of the fray but did allow Ben Stokes to indulge a spell of fast, angry bouncers having employed the tactic with great success of day two.
The Durham all-rounder relentlessly hammered the middle of the pitch, and occasionally a batsman' s body, but could not tempt them into a rash shot. Instead he found himself no-balled by umpire Sundaram Ravi for going too short, too often.
Resuming in the afternoon on 164 for five, Sri Lanka won a minor psychological victory when Root called for DRS on a caught-behind shout against Mendis. There was minimal enthusiasm for the doomed appeal but, as time ticked away, Root gambled in a vain attempt to make a breakthrough.
The game was drifting until, as they have so often this past month, England conjured a magical moment in the field. It was Leach this time, swooping quickly on the on-side as Roshen set off for an ambitious second run. The Somerset man took a moment before scattering the stumps at the non-striker's end with a ferocious, flat throw and was immediately mobbed by his team-mates as Mendis trudged off.
Roshen eked a 30-run stand out of the hyper Niroshan Dickwella but the wicketkeeper's frantic stay ended when he played back to Leach for a regulation bat-pad catch.
Keaton Jennings claimed his sixth catch of the match, equalling the record haul at short-leg, when Moeen got the better of Dilruwan Perera and Moeen ended Roshen's resistance lbw. Correctly sensing a flick of the front pad before contact with the bat, he insisted on DRS to prove his point and bag a fourth wicket of the innings.
Pushpakumara defied England with a sparky 42, merrily carting the spinners with England's reviews all gone, but the fun stopped when Broad used the new ball to pin him on the back of the helmet with a vicious bouncer.
He bravely carried on but Suranga Lakmal could not as he was leg before to the returning Leach in the first over of the evening's play.