But England restricted India to 316 all out, with the bowlers again coming to the fore to take six wickets for 36 runs on the fourth afternoon to halt the hosts' second-innings revival.
Jonathan AgnewBBC cricket correspondent
"After Ahmedabad, a few basic things were rammed home to the England team. Deep down, they knew they had a better bowling attack than India, but that was never going to do them any good unless they had runs to play with. The tourists worked out what they had to do - bat long, sell your wicket dearly."
"It was a great performance level for four and a half days. Everyone performed from one to 11," Cook, on his first tour since being appointed as Andrew Strauss's successor, told BBC Sport.
"To keep India to 300 on that wicket was a fantastic effort and to get six wickets in that session really won us the game. It was a credit to the hard work they've put in that they can perform in these conditions."
The left-hander, in easy reach of the crease at the non-striker's end, moved to avoid Virat Kohli's throw, which then hit the stumps to leave the opener short of his ground.
"I just wasn't thinking and before I knew it I was out," said the Essex man. "It was a frustrating way to go, but if someone offers you 190 before the game you take it.
England's recent record in India
India 1-2 England
India 3-0 England
India 1-0 England
India 1-1 England
India 1-0 England
"The record means a huge amount. I was probably more emotional when I got the 22nd century, but it means a lot to be on top of that list."
Cook's efforts helped England to a lead of 207 and the tourists looked set for an innings victory when India collapsed on the fourth afternoon.
However, Ravichandran Ashwin's 91 not out stretched the game into a fifth day, with England eventually set 41 to win.
They slipped to 8-3 - including Cook, stumped in the first over - but were taken across the line by Nick Compton and Ian Bell.
"It's a great win, especially after losing the toss on a wicket like that," Cook continued. "It's not 'job done' though, there's still five days in Nagpur to go."
Vic MarksEx-England spinner & Test Match Special summariser
"England have beaten India at their own game, despite having lost the toss in their last two matches. They have been playing on subcontinental style pitches, but have emphatically outplayed India. Seven wickets does not really flatter England but that won't matter. It's a terrific turnaround from the first Test."
, who took six wickets in Kolkata, told BBC Sport's Sportsweek programme: "After the first Test we thought it would be tough, but we bounced back last week and to produce an equally impressive performance this week is really special.
"Alastair Cook has been brilliant. He likes to lead from the front and by example and he's clearly doing that at the moment.
"It's vital for the team to get a big first innings score and his 190 was crucial in us doing that here."
We are using archive pictures for this Test because several photo agencies, including Getty Images, have been barred from the ground following a dispute with the Board of Control for Cricket in India, while other agencies have withdrawn their photographers in protest.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.