The hosts took nine wickets after tea on day three - including four for Moeen Ali - as India collapsed to 161 all out and to defeat by an innings and 54 runs.
Mooen's devastating display of spin added to the damage done by man-of-the-match Stuart Broad and James Anderson in India's first innings, with England posting 367 in reply thanks to half-centuries from Joe Root, Jos Buttler and Ian Bell.
And Cook has no doubt it was his bowling line-up which proved the difference as they took a 2-1 lead in the five-match series - even more so with Anderson battling sickness and Broad unable to return to the pitch for India's second innings after being hit in the face by a delivery from Varun Aaron which left him with a suspected broken nose.
"We didn't think we'd get it done today, we thought we'd get enough cricket over the next few days," Cook said. "But it's great to do it in nine wickets in a session. Jimmy hasn't been very well so it's great by him to get out of bed and bowl like that for his nine overs. We knew we had a big effort to bowl for three hours today without Broad but Moeen was fantastic and everything went to plan.
"I've never seen a bloke work so hard and make such a big improvement in such a short space of time. He's really learned how to bowl in international cricket. And Anderson and Broad are a brilliant pairing. I'm very lucky to captain them. In all conditions they've learned how to bowl and they're really good with the younger guys as well.
"Hopefully we can keep them fit for a few more years. The squad will enjoy tonight and a bonus couple of days off and rest up well before going to The Oval."
England went for lunch with a lead of 173 and looked to up the tempo immediately after the resumption. Broad hooked Aaron for successive sixes, but was then hit in the face attempting a third.
He signalled for treatment immediately and, with a bloodied nose and looking groggy, had to retire hurt. He was seen later leaving the ground for hospital.
Cook said: "He's got a bit of a sore nose. It was a really nasty blow, I think he's broken his nose.Fingers crossed there's no cheek fracture and hopefully he'll be all right."
The incident did raise concerns about the way England's batsmen deal with the short ball after Root and Ali were troubled by the same bowler earlier in the match.
Chris Woakes added a quick 26 but England's innings ended on 367 when Anderson fell to his arch-rival Ravrindra Jadeja.
India, without having to face Broad, made a more solid start to their second innings than their first, but it soon unravelled.
Murali Vijay was trapped lbw to become Woakes' first victim of the series and then Gautam Gambhir gloved a legside catch to Buttler off Anderson with the score 53.
Cheteshwar Pujara fell lbw to Ali off the next ball and Ajinkya Rahane chipped a simple catch back to the same bowler soon after. Virat Kohli then edged Anderson to Ian Bell at second slip and when Ravindra Jadeja picked out Chris Jordan in the slips of Ali, India had lost five wickets for 13 runs.
Dhoni's cameo of 27 was ended when Gary Ballance took a fine diving catch at midwicket, giving Ali a fourth wicket. Jordan should have had the eighth wicket Bhuvneshwar Kumar edged him to Bell on a no ball, but the same batsman was run out soon after when he failed to beat an Ali throw while looking for two.
Ravichandran Ashwin clubbed a quick 46, but it mattered little and Jordan completed the match in style. Aaron first gloved him to Buttler and then he bowled Pankaj first ball with an excellent yorker.
England head coach Peter Moores feels his squad are now starting to show their potential. He said: "The last three days have been great for us. The senior players stepping up has been key. The way Stuart Broad and Jimmy Anderson bowled up front in the first innings set a tone for us.
"Also, the young players have started to step up. International sport gives you a chance to develop quickly and Mooen has taken that chance. He's worked very hard in the nets and he's starting to show that in the middle."