Rahul and Pujara were out for 13 and 6 respectively after India were put in to bat on a Sabina Park pitch which had some grass on it.
Captain Virat Kohli and the other opener Mayank Agarwal were batting at 41 off 99 balls and 5 off 30 deliveries respectively at the lunch break.
After West Indies captain Jason Holder opted to field, Rahul and Agarwal negotiated the first half an hour without taking any risk on a tricky pitch.
Opening bowlers Kemar Roach and Shannon Gabriel, however, could not get much movement from the pitch.
Holder introduced himself in the seventh over and immediately got the breakthrough by having the wicket of Rahul to break the 32-run opening stand.
It was a peach of a delivery which jagged away from middle stump with Rahul edging it to Rakheem Cornwall at first slip for a comfortable catch. Rahul hit two fours in his 26-ball innings.
Holder was getting the bounce and movement and debutant Cornwall made the Indians toil for runs with his tight off-spin bowling.
India were 36 for 1 in 13 overs at the drinks break.
Pujara, who came after Rahul's departure in the seventh over, struggled again as he took 15 deliveries to open his account and was finally out for 6 which came from 25 balls.
The bounce extracted by 26-year-old off-spinner Cornwall, who stands at 6ft and 5 inches and weighs more than 140kg, did bring Pujara's downfall as he hit straight to the hands of Shamarh Brooks.
India were 46 for 2 in the 17th over at the fall of Pujara but Kohli and Agarwal ensured that the visiting side doesn't lose any further wicket. The duo added 26 runs in 13.1 overs for the unbroken third wicket.
Agarwal has so far hit five fours from the 99 balls, while Kohli's 5 came from 30 deliveries.