Left to negotiate 13 testing overs from the tourists before the close of play, West Indies lost openers Kraigg Brathwaite and John Campbell.
The hat-trick ball involved some drama as it required a review from the fielding side to overturn the original not out verdict.
Virat Kohli led a workmanlike effort from India's top order with a carefully crafted innings of 76.
Keemo Paul will replace fast bowler Miguel Cummins, who went wicketless in the first Test at Antigua conceding 69 runs in 20 overs.
The hosts were demolished in 26.5 overs for their lowest Test innings total ever against India with Bumrah returning the outstanding figures of 5-7.
Rahane, who top-scored with 81 in his team's first innings total of 297, will resume on the fourth morning on 53 alongside Kohli on 51.
Cummins fell six minutes shy of breaking the dubious feat of New Zealand's Geoff Allott who lasted 101 minutes before being dismissed for nought.
A late burst by senior fast bowler Ishant Sharma earned him a five-wicket haul and reduced the West Indies to 189 for eight.