Kraigg Brathwaite, who is standing as captain after Jason Holder suffered a last-minute ankle injury, juggled his bowlers who often looked toothless.
Shaw is the seventh youngest global batsman to score his first Test century. Subsequently, Shaw offered an easy return catch to leggie Devendra Bishoo trying to play against the turn.
The last session saw India maintain a good run rate with Kohli and Rahane (41) stitching a 105-run partnership and helping India get past the 300-run mark.
He was not facing one of the best attacks but he gave ample evidence of his brilliance on way to a memorable hundred.
Pujara also played an uncharacteristically quick knock as he brought up fifty off 67 deliveries, but missed out on his hundred after being caught behind off debutant paceman Sherman Lewis.
The Australian received a 12-month ban for his part in the ball tampering controversy during the Cape Town Test and can not officially return to playing state, Big Bash and national level cricket until March 29 next year. Shami reaches for the ball outside off and it sails over the second slip. Rahul was better off not taking the DRS as the ball was clearly crashing into the stumps.
At the break he was batting alongside Cheteshwar Pujara, on 56, in a strong recovery after India lost an early wicket in Rajkot.
The teenage prodigy collected two fours off Bishoo in the 11th over and did the same to Chase in the 20th over.
Their lead pacer, Kemar Roach, was already ruled out of the series opener due to a bereavement in the family.