Pant scored 109 off 128 balls for his sixth Test century as India declared their second innings at 4-287 off 64 overs in Chennai.
Shubman Gill notched his fifth Test hundred, scoring an unbeaten 119 as India set the tourists a 515-run target.
In reply, Bangladesh lost four wickets after the tea break, with skipper Najmul Hossain Shanto holding ground with 51 not out off 60 balls. Shakib al Hasan (5 not out) gave him company as play was stopped owing to bad light with Bangladesh needing a further 357 runs.
Earlier, Pant and Gill had put on 167 runs for the fourth wicket, including 124 in the wicketless morning session.
Pant reached 50 off 88 balls, while Gill got there off 79 deliveries, as India went to lunch at 3-205.
After lunch, Pant equalled MS Dhoni's record for most Test hundreds by an Indian keeper-batter. Dhoni also scored six hundreds in 90 Tests, with Pant reaching the landmark in only his 34th Test.
Pant suffered career-threatening injuries in a car crash in December 2022, last playing Test cricket against Bangladesh earlier that month. He was away from the game for 17 months recuperating from knee and back injuries, before returning for the 2024 Indian Premier League.
He was also part of India's T20 World Cup campaign.
Pant reached 100 off 124 balls, and hit 13 fours as well as four sixes. Gill scored his ton off 161 balls.
Pant was dismissed after his century, out caught off Mehidy Hasan Miraz (2-103), while Lokesh Rahul made 22 not out before the declaration.
Bangladesh went to tea at 0-56 in 13 overs, with openers Zakir Hasan (33) and Shadman Islam (35) putting on the highest opening partnership for the visitors on Indian soil.
Until then, only one wicket fell in two sessions, compared with 23 wickets across the first two days.
After tea, Jasprit Bumrah got the much-needed breakthrough, with Hasan caught at gully by Yashasvi Jaiswal.
Ravichandran Ashwin then picked 3-63 as he ran through the Bangladesh top order. He troubled the batters with bounce off the surface and induced the batters into false shots.
Shadman was out caught at midwicket, with Gill taking a diving catch. Mushfiqur Rahim was dismissed for 13, again to a brilliant forward diving catch by Rahul at mid-on.
In between, Ashwin bowled Mominul Haque for 13 as Bangladesh were reduced to 4-146.
Skipper Shanto was the only batter who provided resistance, reaching his fourth Test half-century off 55 balls. He hit four fours and three sixes.
Play was then called off later in the session because of bad light.
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