Playing his first test in 19 months, he took five for 33 in 23.4 overs as West Indies lost their last seven wickets for 23 runs to be all out for 225 in the first innings.
India then reached 157 for three by the close of play, 285 runs ahead, leaving captain Virat Kohli to ponder the best time to declare on the final day on Saturday.
All of Kumar's dismissals were achieved with quality deliveries, three caught in the slips, one bowled and one lbw.
"That was a spell of consistent, high-quality swing bowling," said former West Indies player Jeffrey Dujon on television commentary.
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"He gave all the batsmen trouble, swung it in, swung it out, set up Marlon Samuels beautifully and really didn't move too much from that tight off-stump line."
Samuels was perhaps Kumar's most prized scalp, out for 48 when he was beaten by an inswinger that he chopped on to his stumps.
One run earlier Jermaine Blackwood was caught at second slip off Kumar for 20. Opener Kraigg Brathwaite top-scored for West Indies with 64.
Kohli went cheaply in India's second innings, trapped lbw by Miguel Cummins for four after making just three in the first knock.
Ajinkya Rahane (51) and Rohit Sharma (41) were unbeaten at stumps.
India lead the four-test series 1-0.
(Reporting by Andrew Both; Editing by Tony Jimenez)
