James Anderson displayed his reverse-swing mastery with a three-wicket burst in the morning session to effectively snuff out India’s hopes of a draw while left-arm spinner Leach (4-76) then ran through the lower.
The hosts, who had been set a daunting 420-run victory target, got a fighting 72 from captain Virat Kohli while opener Shubman Gill scored an elegant 50 but they were bowled out for 192 in the second session of the final day.
The tourists lead the four-match series 1-0, with the second Test set to start on Saturday, again in Chennai.
Root, who smashed 218 in the first innings, said batting first had been key.
“It was a crucial toss and we posted a very good score and to take 20 wickets in alien conditions … huge credit to the bowlers,” he added.
“That’s got to be the way we have to win out there — bat for long periods for time in the first innings.”
England have now recorded six straight wins in away Tests, including triumphs in South Africa and Sri Lanka.
The win also boosted England’s chance of reaching the final of the inaugural World Test Championship in June on home soil.
Root’s team need to win two more Tests against Kohli’s men to book a place against New Zealand, while India must prevail by a 2-1 margin to pip England.
India skipper Kohli said there was lots of room for improvement from his side.
“England played far more professionally,” he added. “You need your bowling unit to step up and perform more collectively.
“We are a side that’s always looking to improve. We know how to bounce back strongly, and we’re going to give our best effort in the next game and start off on the right note.”
Earlier on Tuesday, Leach had given England the perfect start by removing Cheteshwar Pujara after the hosts had resumed on 39-1. Pujara, whose dour batting could have helped India bat out the final day and secure a draw, made 15.
The flamboyant Gill brought up his third fifty in only his fourth Test before Anderson struck twice in four balls. The 38-year-old breached Gill’s defence to dismiss him with a hint of reverse swing.
Next man in Ajinkya Rahane survived a strong lbw appeal but it did not matter as Anderson’s next delivery sent his off-stump cartwheeling away.
The dangerous Rishabh Pant became Anderson’s third victim, offering a leading edge to Root at short cover.
Kohli and Ravichandran Ashwin resisted England for nearly 18 overs, during which Ashwin sustained blows on his wrist and was smacked on top of his helmet by Jofra Archer.
Leach dismissed Ashwin, who batted more than an hour for his nine runs, but India’s resistance effectively ended when Stokes bowled Kohli.