Chasing 399 for victory, England were 95-1 before they lost five wickets in the morning session to allow India to take charge of the contest.
Opener Zak Crawley struck 73, while Ben Foakes and Tom Hartley produced defiant 36s down the order but England were eventually all out for 292 in the second session on day four.
Left-arm spinner Axar Patel removed Rehan Ahmed, who did a decent job as England's nightwatchman by scoring a brisk 23.
Ashwin then rattled England by dismissing Ollie Pope and Joe Root in successive overs.
Pope made 23 before edging the off-spinner and the ball stuck 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 hit Ashwin and ended up offering a top-edge to Patel at backward point.
"We know winning a test match in these conditions is not going to be easy, but the bowlers stepped up," said India captain 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."
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.
Skipper Stokes said he had "full belief" that they could chase the total down.
"The way we have gone about taking on challenges like that is what we are about," he said.
"In moments like that, when you have scoreboard pressure, that is when we get the best out of ourselves as individuals," added Stokes.
"The way we applied ourselves and put India under pressure was great, unfortunately we didn't end up on the right side of the result."
The third test in Rajkot begins on February 15.
