Tagenarine Chanderpaul (24) and Jermaine Blackwood (20) will resume on Monday with West Indies still 289 behind and needing an extraordinary batting effort on a fifth day track to escape defeat.
Earlier, leading an inexperienced Indian pace attack, Mohammed Siraj claimed 5-60 to help bowl out West Indies for 255 after the hosts resumed their first innings on 229-5 at the Queen's Park Oval in Port of Spain.
West Indies lost their last five wickets for only 26 runs with Siraj wreaking havoc with the ball.
Possessing a handy lead of 183, Indian batters came out all guns blazing in their second innings scoring as if playing a T20 match.
Skipper Rohit Sharma (57) raced to a 35-ball fifty as he and Yashasvi Jaiswal (38) took India to the 100-mark in 12.2 overs.
Promoted to number four, Ishan Kishan struck an unbeaten 52 off 34 balls, his maiden test fifty in his second test, as India racked up 181-2 in 24 overs before declaring their innings.
Off-spinner Ravichandran Ashwin further cemented India's position on a rain-marred with two late blows.
West Indies captain Kraigg Brathwaite made 28 before top edging Ashwin at short fine leg, while Kirk McKenzie was trapped lbw for a duck.
For India, Siraj particularly stood out for the way he led the attack in the absence of a recovering Jasprit Bumrah and a rested Mohammed Shami.
"I relish responsibilities," the 29-year-old told reporters.
"To step in on the field with a responsibility on my shoulders gives me extra motivation and it is challenging as well and I like challenges.
"It was hot and humid and because of the rain, we were going on and off the field. It's difficult for a fast bowler to keep yourself charged up."