Left-hander Jaiswal batted through the day in his first test innings for 143 not out and India finished the day with a first-innings lead of 162 on 312 for two after adding 232 runs to their overnight tally for the loss of only two wickets.
Rohit secured his 10th test century with a four but was dismissed caught behind for 103 off the spin of debutant all-rounder Alick Athanaze from the following delivery.
Athanaze was one of eight bowlers used by West Indies on the slow pitch in the capital of Dominica.
Shubman Gill followed soon afterwards when Jomel Warrican had him caught by Athanaze in the slips for six but Jaiswal batted on in a partnership with Virat Kohli (36 not out) that was worth 72 runs at the close of play.
Jaiswal, who had faced 350 balls at stumps, was the 17th Indian to score a century on debut.
India, who dismissed West Indies for 150 on day one with the help of a five-wicket haul from Ravichandran Ashwin, will be looking to build a sufficient lead on day three to be in a position to win the test without having to bat a second time.
The second and final test in the series will take place in Port of Spain from July 20 before the teams face off in three one-day internationals and five Twenty20 matches.
