India extended their overnight score to 416 all out, with Hanuma Vihari scoring his maiden test century, before reducing the hosts to 87-7 at stumps on day two, still 329 runs behind.
Bumrah dismissed Darren Bravo (4), Shamarh Brooks (0) and Roston Chase (0) in successive balls in the ninth over of West Indies' first innings. Bumrah's hat-trick came after a successful lbw review against Chase, with captain Virat Kohli challenging the initial not out decision on an lbw appeal.
Harbhajan Singh and Irfan Pathan also have test hat-tricks for India.
Only Shimron Hetmyer (34) and captain Jason Holder (18) offered resistance for the hosts, sharing a 45-run stand for the sixth wicket after West Indies had been reduced to 22-5.
India resumed on 264-5 on day two in the final test after being put into bat on day one. Vihari went on from 42 not out to score 111 in his debut century and Ishant Sharma hit 57 as India posted a convincing first innings total that had looked in doubt after Holder bowled Rishabh Pant (27) with the first ball of the day.
Ravindra Jadeja added 16 before he was caught by Darren Bravo off Rahkeem Cornwall just before lunch to reduce India to 302-7. Vihari and Sharma then combined for a 112-run stand for the eighth wicket with both reaching their highest test scores midway through the middle session.
Vihari's hundred arrived after a sharp single in the 133rd over and the batsman celebrated his milestone by punching the air and raising his bat before embracing Sharma. Carlos Braithwaite ended the rally soon after by snaring Sharma for 57 and Mohammed Shami was out next over to Cornwall without scoring.
Vihari was the final wicket to fall after he holed out to Kemar Roach off Holder, who finished with 5-77 off 32.1 overs.
India won the first test by 318 runs. It was its biggest away win by runs in test cricket, and Kohli's 27th win from 47 tests as India captain, equaling M.S. Dhoni's record for wins at the helm of the national team.