This result also completes a 100 percent record for the tourists in all completed matches played on this tour.
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.
Chris Gayle, Colin Munro, Daren Sammy, Munaf Patel among overseas stars for LPL auction
Sarfaraz Ahmed, Shan Masood, Haris Sohail among captains named for National T20 Cup
Galle Gladiators sign Shahid Afridi as icon player for Lanka Premier League
PSL 5 to return on November 14; all matches in Lahore
PSL Schedule Live Streaming and PSL 5 Matches Details
National T20 Cup: Salman Butt declines to lead Central Punjab second XI
Dawid Malan dethrones Babar Azam to become top-ranked T20I batsman
PCB finalises three-year broadcast deal with PTV
PCB announces major salary hike for domestic cricketers in new pay structure
'Team was already on a losing streak when I took over' - Misbah on Pakistan's T20I decline