James Anderson got the key wickets of Gill, Rahane and Pant on Day 5 at Chepauk.
On Day 5, Anderson got the ball to reverse and bowled just about the right lengths. He dismissed Shubman Gill (50), Ajinkya Rahane (0) and Rishabh Pant (11) and broke the back of Indian middle-order.
The hosts eventually got bowled out for 192 while chasing the daunting target of 420 in the fourth innings.
"It came out well. The ball was reversing. We knew we had to get the lengths right and I managed to do that," Anderson said after the match.
"I got a bit of luck with the bounce there. Reverse swing was huge for us. The pitch was obviously slow and deteriorating so movement in the air makes us seamers feel we can get a wicket any ball," he added.
Anderson, who picked up five wickets in the match, said it was a hard grind for five days but was happy to have emerged victorious.
"It was hard graft out there for five days. I am feeling great. Had a good tour of Sri Lanka and managed to carry on the form here as well," the 38-year-old said.
"Glad to have got the nod for this game, we have a good lot of bowlers here. Now I need to rest well for three days and come hard again."
Joe Root led his team from the front as he slammed 218 to propel the visitors to a massive 578 in the first innings. The England captain also scored a quick-fire 40 in the second essay, enabling the tourists to set an improbable target for India.
Root was deservedly adjudged Player-of-the-Match and Anderson showered rich praise on his skipper.
"Root is a great player of spin. Has led us from the front, led by example. Hopefully, he can continue the form and the rest of us can chip in," he concluded.
(With PTI inputs)