Jennings, who has replaced the injured Haseeb Hameed in the squad, and one-down Joe Roots were at the crease during the break as England piled up 117-1 at lunch on the opening day here.
The visitors' captain, after looking comfortable in his stay, suddenly had a rush of blood and stepped out injudiciously to the third ball from Ravindra Jadeja and was easily stumped, with 15 minutes remaining for lunch, with the tourists' total reading 99.
Jennings, who had forced his way into the squad on the strength of scoring over 1500 runs in domestic cricket, was lucky on two occasions within the first half hour, when he could have been dismissed.
On the first occasion he fended at a short ball from Umesh Yadav but the ballooning ball could not be latched on to by gully fielder Karun Nair with his left hand despite a leap. The batsman was then yet to open his account after having faced nine balls.
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The decision was reviewed and the batsman escaped as the ball had hit the pad in line with off, just marginally, and the original decision of the umpire stayed.
Slowly growing in confidence, he struck the same bowler for two fours in successive balls with authentic shots to help England end the first hour with 49 on board.
Cook, at the other end, played cautiously and took singles with nudges into the vacant areas while hitting fours off a few loose balls.
Ashwin had Cook in trouble in his last over, but Cheteshwar Pujara was not alert enough at short leg to catch a leading edge when the opener tried to turn the off spinner to the leg. The opener was then on 39.
Luckily for the home team the miss did not prove too costly as Jadeja replaced Ashwin and got rid of Cook for his and India's first wicket. The batsman was out to the 60th ball he faced in his 103-minute stay and essayed five fours.
Earlier, India made two changes to their playing eleven, by including Bhuvneshwar Kumar and KL Rahul into the playing eleven.
England also made two changes to their playing XI, handing Test debut to Jennings and including rookie Jake Ball as the fourth pace bowler in his second Test.
England are trailing India by 0-2 and facing a must-win situation to keep the five-Test series alive.