Fast bowlers Umesh Yadav and Mohammed Shami took four wickets each as the visitors bowled out West Indies for 243 in their first innings.
After India captain Virat Kohli enforced the follow-on, the hosts were then 21-1 at stumps, trailing by 302 runs, and with two days left perhaps only rain can save them.
Shami and Yadav led a strong five-pronged attack at the Vivian Richards Stadium in North Sound.
Yadav (4-41) and Shami (4-66) found surprising life in the pitch in the first innings while Wriddhiman Saha behind the stumps featured in six dismissals (five caught and one stumped), equalling the record by an Indian wicketkeeper.
India were aided in their quest by batting that retired West Indies great Richards, in the commentary box, described as “very, very ordinary”.
Opener Kraigg Brathwaite top-scored for the home side with a disciplined 74 off 218 balls in nearly five patient hours at the crease.
Number eight Shane Dowrich also hit a lively 57 not out and put on 69 for the eighth wicket with captain Jason Holder (36).
Marlon Samuels and Carlos Brathwaite, heroes of April’s World Twenty20 title triumph, went cheaply.
Samuels was caught behind for one while Brathwaite made an embarrassing golden duck when he did not play at a Yadav delivery and watched in dismay as his off stump was knocked over.
“India really bowled well today,” Kraigg Brathwaite said in an on-field interview. “With three (fast bowlers) and spinners they had a lot to choose from.
“It was never easy because they really stuck to their plans. It was a tough day of test cricket.”
India relentlessly pressed home their advantage late in the day as West Indies faced 13 overs in their second innings.
Kraigg Brathwaite could not emulate his first-innings knock, trapped lbw for two by paceman Ishant Sharma.
Darren Bravo (10) and Rajindra Chandrika (9) then survived until stumps.