Cook showed glimpses of returning back to form as he remained unbeaten on 48 off 96 balls after Ravindra Jadeja dropped a dolly at third slip denying debutant Pankaj Singh his maiden Test wicket.
Cook hit five boundaries and is batting in company of an in-form Gary Ballance (4 batting).
Jadeja however made partial amends as he held onto a comparatively difficult catch off Mohammed Shami's bowling to dismiss Sam Robson (26) as he pushed hard at a delivery without much feet movement.
Rohit Sharma also predictably came into the side in place of Stuart Binny, who did precious little in the second Test at the Lord's.
More From This Section
For England, Jos Buttler made his debut replacing Matt Prior behind the stumps, and Chris Woakes and Chris Jordan were brought in place of Liam Plunkett and Ben Stokes.
The Indian bowlers didn't really make the ball talk on a pitch that didn't have the kind of juice that was there in the Lord's pitch.
To his credit, Cook tightened up his defence further after being given a 'life' and didn't give another chance to the Indian bowlers, who fared better during the later part.
The 50-partnership for the opening pair came up in the 19th over, when Cook played an immaculate cut shot for four. This was only his second four over nearly 90 minutes of batting, and 50 balls after his first boundary, showing how hard he was trying in the middle.
But then India did break through as Shami returned in the 21st over to snap up Sam Robson's wicket. He edged a marginal out-swinger from back of length to third slip where Jadeja redeemed himself this time around. Robson scored 26 runs off 59 balls with 4 fours and added 55 runs for the first wicket.