Kohli was on 111 at stumps, his fifth 50-plus score in the last seven test innings, helping put India, currently the top-ranked test team, on course for a big first innings total.
Ajinkya Rahane, returning from a finger injury, was batting on 45 at the other end, having played second fiddle to his skipper in their 122-run fourth wicket partnership.
Playing their first test on Indian soil, Bangladesh got an early breakthrough when the sprightly Taskin Ahmed dismissed Rahul for two with the fourth delivery of the match.
While others might have left the fuller delivery well outside the off-stump, Rahul went for an expansive cover drive without moving his feet, only to drag the ball onto his stumps.
Vijay and Pujara shrugged off occasional discomforts to help India overcome the loss with their fifth century stand of the current home season.
Off-spinner Mehedi Hasan induced a couple of edges but could not separate the batsmen despite some occasionally comic running between the wickets.
In one such mix-up, both batsmen found themselves at the same end. But Mehedi could not collect a throw and dislodge the bails at the non-striker's end which could have sent Vijay, then on 35, back to the pavilion.
Vijay brought up his fifty in 82 balls and went on to hit Shakib Al Hasan straight back for the only six of his knock which also contained 12 boundaries.
Pujara edged Mehedi to Bangladesh captain Mushfiqur Rahim after a composed 83, and left-arm spinner Taijul Islam bowled Vijay around the leg after the break at tea.
There was no real respite for the tourists, however, as Kohli scored freely against the limited bowling resources of a team ranked eight places below India.
The right-hander needed 130 balls to bring up his fourth century in seven tests, reaching the mark with his 10th boundary.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Sudipto Ganguly and Hugh Lawson)
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