Chasing a victory target of 399, England were 95-1 before losing five wickets in the morning session as India set about levelling the series.
Opener Zak Crawley struck 73, with Ben Foakes (36) and Tom Hartley (36) producing defiant knocks down the order, but England were eventually dismissed for 292 in the second session on day four.
"We know winning a test match in these conditions is not going to be easy, but the bowlers stepped up," said India captain Rohit Sharma.
"The wicket was good to bat on, a lot of the batters got starts but didn't convert. This win gives us a lot of confidence. Very proud of such a young squad."
Former captain Virat Kohli missed the first two tests for personal reasons and India were also without batter KL Rahul and all-rounder Ravindra Jadeja after they picked up injuries in the series opener in Hyderabad.
Left-arm spinner Axar Patel removed Rehan Ahmed, who did a decent job as England's nightwatchman by scoring a brisk 23 after play resumed on day four.
Ashwin then rattled England by dismissing Ollie Pope and Joe Root in successive overs.
Pope made 23 before edging to Rohit Sharma's outstretched left hand at slip.
Root, nursing a finger injury, smashed Patel for a six but gifted his wicket when he danced down the track to Ashwin and top-edged a catch to Patel at backward point.
Crawley batted with characteristic aggression and Jonny Bairstow (26) also looked set before they departed in successive overs before lunch.
England's hopes was effectively snuffed out when Ben Stokes was run out for 11.
Foakes and Hartley refused to throw in the towel, though, prompting Rohit to bring back pace spearhead Bumrah.
Bumrah had Foakes caught and bowled with a slower delivery before Hartley had his stumps rearranged by a Bumrah outswinger.
"We had full belief we could chase down 399," Stokes said afterwards. "The way we applied ourselves and tried to put the India bowling attack under pressure was great."
Bumrah's nine-wicket match haul in a spin-dominated contest earned him the player-of-the-match award.
The third test in Rajkot begins on February 15.