Jennings had seven partners in total, Ben Stokes the most productive with a bustling 62, while Ben Foakes' late cameo saw him clearing the ropes three times - showing off yet another string to his bow on his remarkable debut.
England needed five wickets in the last session to go 1-0 up in the three-test series, and there was not much resistance from Sri Lanka.
Sri Lanka began the day on 15/0 - 447 runs away from their victory target - and did well to progress to 51 before Leach delivered England their first wicket of the day in the 23rd over, pinning Kaushal Silva lbw on the sweep for 30.
Root reserved special praise for Foakes, who stole the show with 107 runs in his first innings on the Test stage.
Jennings gritted his teeth and held his end up, happy to nudge his side towards a winning position rather than attempt anything outside of his comfort zone.
From 98/3 at lunch, Sri Lanka lost two wickets in the afternoon to Leach. Stokes shrugged off his early tentativeness to clobber three sixes in his 62 before Perara spun one past his bat to hit the top of the off-stump with a handsome delivery.
But shortly before tea he became Perera's second scalp, a demon of a ball pitching outside leg and angling back to take the Durham all-rounder's off stump. You can go to him and he has given really good advice.
Last man out was silver fox Rangana Herath, cricket's most successful left-arm spinner in history, in his last match before retirement. "You can not come to worldwide cricket and learn that. I´ve had to develop myself", Jennings said.
"When you´re waking up at 6.30 in the morning and reading about your technical deficiencies it´s not human to say it wouldn´t affect you", he added.
Sri Lanka´s spin bowling coach Piyal Wijetunge said however that the hosts still have a chance to win, in part because the notorious Galle wicket is behaving differently this time. Never before had an English cricket team won the opening match of a Test series in Sri Lanka.
"Well played fantastic and thoroughly deserved by the hard work".
"There is a big challenge, but it´s not the kind of wicket we usually see in Galle".
"We need to regroup as a team".
"One of the best weeks of my career so far, if not the best".