Any hopes West Indies had of finding further immediate breakthroughs were then dashed as Shaw and Pujara combined in a 206-run second wicket stand to consolidate the innings.
Chesteshwar Pujara added 86 and captain Virat Kohli was 71 not out, as India notched its highest test total in a day's play against the West Indies.
India were 364 for four at stumps after electing to bat first.
Shaw led India's fightback after they lost opener Lokesh Rahul for nought in the very first over off Shannon Gabriel.
He raced into the 90s with another pulled boundary off the ineffective Bishoo before raising three figures off just 99 balls almost three-quarters of an hour after the interval, punching Paul to cover for a couple.
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Shaw reached the landmark with two runs off paceman Keemo Paul, punching his fist in the air as the home crowd gave him a standing ovation.
Prithvi Shaw scored at better than a run a ball before lunch on his test debut in Rajkot. I used to play 30-35 school games in a year, and played a lot of (first-class) Ranji (Trophy) cricket as well. Holder injured his ankle during the preparatory camp in Dubai, and in his absence, opener Kraigg Brathwaite was handed over the captaincy for the second time in Tests. Roach will be absent as he has flown back to attend a family issue and the lack of top-quality spinners is a cause for concern.
He built a solid partnership with Cheteshwar Pujara.
While India is set to go with five bowlers, questions remain whether the Rajkot wicket-which has the history of offering bucketful of runs-will help the bowlers much.
Kohli, who is the world's number one Test batsman, got to his half-century against Bishoo and was getting more risky with each passing ball.
Rahane, too, lacked the ease with which Shaw batted earlier in the day, and fell on 41 to Roston Chase in the last hour.