The Windies lost three quick wickets in the last few overs of Day 2 to leave them reeling at 189/8.
India were bowled out earlier in the day for 297, with Ravindra Jadeja hitting a valuable 58, batting at number 8.
In reply, the Caribbean boys got a good start from openers Kraigg Brathwaite and John Campbell, but soon lost both openers and newcomer to the crease Shamarh Brooks to be at 50/3.
It left Darren Bravo and Roston Chase to rebuild the innings, but Bravo was trapped LBW by Bumrah for 18 with the score on 88/4.
Roston Chase (48) then fell 42 runs later, with hosts in trouble at 130/5.
Shai Hope and Shimron Hetmyer were eyeing the end of the day’s play, but a flurry of wickets left the Windies in some serious trouble.
Hope fell caught behind by Sharma for 24 (174/6) in the 54th over.
Sharma then struck two overs later, getting Hetmyer caught and bowled for 35 (179/7), and three balls later Kemar Roach for a duck (179/8).
Jason Holder (10) and Miguel Cummins (0) saw out the end of the day’s play, with the West Indies trailing by 108 runs with 2 wickets remaining.
Ishant Sharma has so far been the pick of the bowlers taking 5 wickets.