Jennings, who replaced injured Haseeb Hameed, showed no nerves and took full advantage of a dropped catch, to remain unconquered on 103 in England's total of 196 for two at tea.
Jennings became only the 13th batsman from his adopted country to make a hundred on debut as the tourists moved on from their lunch score of 117 for one by adding 79 runsin the middle session.
Root was sent back by Ravichandran Ashwin through a smart catch at slip by home team captain Virat Kohli after Cook had been stumped by stumper Parthiv Patel off left-arm spinner Ravindra Jadeja.
At tea, the unbroken third-wicket partnership between the tall Johannesburg-born left-hander Jennings, son of renowned coach Trevor, and Ali, was worth 60 in 142 balls.
Jennings played some audacious reverse sweeps too, including when on the cusp of his landmark, to reach the hundred in 186 balls and laced with 12 fours.
Ashwin, changing ends, bowled a fine 11-over second spell for 21 by extracting turn and bounce and also got rid of danger-man Root as the batsman edged a ball and Kohli moved to his left to pick up the low catch.
Jadeja kept it tight at the other end in an eight-over spell on either side of lunch that also gave India the breakthrough before the break when Cook had a rush of blood and came out to slog the left arm spinner and was easily stumped.
Just before the end of the third hour's play umpire Paul Reiffel, standing at square leg, had to leave the field off freak injury, hit on the back of the head by a lobbed throw from deep square leg by Bhuvneshwar Kumar.
The incident happened in the 49th over and third umpire Marias Erasmus took over Reiffel's duties, while C Shamsuddin took over the latter's position indoors.
(REOPENS DEL 22)
Jennings kept his cool and reached his maiden hundred by reverse sweeping Jayant for a four. He had struck the same bowler for a four in similar fashion earlier on too.
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 yet to open his account after having faced nine balls.
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 while hitting fours off a few loose balls.
Ashwin had Cook in trouble in his last over before lunch, but Cheteshwar Pujara was not alert enough at short leg to catch him when the batsman was on 39, a leading edge as the opener tried to turn the off spinner to the leg.
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, handing debut to Jennings and including rookie Jake Ball as the fourth paceman.
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