India captain Virat Kohli and Windies' stand in skipper Kraigg Braithwaite.
Shaw is the seventh youngest worldwide batsman to score his first Test century. He did not take many chances and that allowed Rishabh Pant (92 off 84 deliveries) to play freely on the other end as India marched towards a massive total in the first innings.
Already set back by the loss of seasoned new-ball bowler Kemar Roach who returned to Barbados last week due to the death of his grandmother, West Indies suffered yet another blow when captain Jason Holder was ruled out of the match with an ankle injury.
Shaw, who is 18 years and 329 days old, reached his hundred off just 99 deliveries in the second session on day one.
The teenager also scored the fourth-fastest fifty by an Indian on Test debut.
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Brief scores: India 364/4 (Prithvi Shaw 134, Cheteshwar Pujara 86, Shannon Gabriel 1/66) against West Indies on Day 1.
Shaw and Pujara added 206 runs for the second wicket as the tourists were sent on a leatherhunt for large parts of the second session.
The Indian skipper also surpassed Australia's Steve Smith to go second in the list of centurions among active players, behind South Africa's Hashim Amla on 28.
Despite Kohli scoring nearly 600 runs - and two centuries - India suffered a 4-1 defeat to England in the summer.
Shaw's hundred is a massive one for India in two respects - one, it breaks the kind of monopoly that the Vijay-Dhawan pair enjoyed and two, it shows that India can benefit from moving on from tried and tested options.
He reached his 20th half-century in Tests off 100 balls and all told, has so far counted four fours in a 137-ball innings.