The Indian lower-order contributed 213 runs to the total score in their first innings after the hosts were reduced to 204-6 following a fightback by the England bowlers on the second day.
The remaining two Tests provide more opportunity to Kohli, who had led India to victory in the ICC U19 Cricket World Cup in Malaysia in 2008.
Hameed and Woakes then added 43 runs between them to take the visitors near the 200-run mark before Shami produced some excellent short pitched bowling with the second new ball to hasten England's demise.
To rub it in on day three, England were treated to the sight of the home side's lower order batting more sensibly than their own top order.
England were bowled out for 236 in their second innings earlier on day four with Joe Root leading the way with a defiant 78.
Resuming on 78-4, England suffered early setbacks.
Ashwin's wicket ended a 97-run seventh-wicket stand between him and Jadeja. He smacked Jadeja for a big six over long-off, and continued his aggressive batting with another boundary.
Root batted with goal to register his 25th Test fifty but his gritty knock came to end just 10 minutes before the break when Jadeja dismissed him for 78.
Hameed refused to give up, however, defying the Indian bowlers with solid defence alongside Chris Woakes who made a fluent 30.
In one brilliant over from Mohammed Shami, both Woakes, who fended the fast bowler behind one delivery after his stem guard was dislodged after he was hit on the helmet, and Adil Rashid, hooking another bouncer to fine leg, departed.
Anderson and Hameed stuck together for an annoying 10th wicket partnership before Anderson foolishly ran himself out for 5 leaving Hameed high and dry on 59 not out.
Root was ready to play in occasional ones and twos even though Jadeja tried to hurry him which forced the Yorkshire man to play the left-armer off his backfoot.
Cook said that Hameed was having difficulty in holding the bat. It was good accurate bowling as you expect from India in these conditions.
"I looked at that wicket along with a number of people before the game that we thought was dry".
England'sJonny Bairstow also reached a career-best ranking of ninth as he has shot up three places with knocks of 89 and 15, while Australia opener Usman Khawaja moved up nine places to 19th rank after a match-winning knock of 145 that helped Australia win the final Test against South Africa by seven wickets.
The action now shifts to Mumbai with the fourth Test scheduled to start on December 8.