Tendulkar was unbeaten on 33 as India reached 152-2 in their second innings at stumps on the third day after being set a challenging 276-run target to win the match on a low, slow pitch.
India now need 124 more runs to win the match with eight wickets in hand.
Tendulkar, needing just one more century to score an unprecedented 100 international hundreds, completed 15,000 runs during his unbeaten 87-ball knock when he took a single off leg-spinner Devendra Bishoo.
The master batsman, the world's leading scorer in both Test and one-day cricket, looked at the sky and raised his bat before being congratulated by team-mate Rahul Dravid (30 not out) and the West Indies players.
The West Indies were bowled out for 180 in their second innings, with off-spinner Ashwin bagging 6-47.
Ashwin, who took 3-81 in the first innings, finished the match with 9-128, the second-best match haul by an Indian debutant after leg-spinner Narendra Hirwani's 16-136 against the West Indies at Chennai in 1988.
India got off to a brisk start as Virender Sehwag (55) and Gautam Gambhir (22) added 51 runs in 9.5 overs for the opening wicket before both were dismissed in the last session when looking well set.
Gambhir fell leg-before to part-time off-spinner Marlon Samuels, but Sehwag continued to dominate the West Indies attack with aggressive shots before playing seamer Darren Sammy's delivery on to his stumps.
Sehwag, dropped on 12 when paceman Ravi Rampaul failed to hold on to a return catch, hit two sixes and five fours in his run-a-ball knock.
The West Indies were earlier reeling at 84-7 before managing to set a challenging target as they added 96 runs for their last three wickets, thanks to veteran Shivnarine Chanderpaul (47) and skipper Sammy (42).
Chanderpaul, who cracked an impressive century in the first innings, and Sammy both counter-attacked to add 40 useful runs for the eighth wicket before falling to Ashwin in the space of six overs.
Chanderpaul was trapped leg-before after hitting seven fours in his 58-ball knock while Sammy was bowled off a delivery that kept low.
Sammy played some attractive shots during his 37-ball knock, once smashing fast bowler Ishant Sharma for a straight six.
The Indian bowlers called the shots in the morning session when they captured five wickets for 88 runs after the West Indies had resumed at 21-2, with Ashwin and debutant seamer Umesh Yadav (2-36) doing the maximum damage.
Ashwin trapped Darren Bravo leg-before for 12 runs and then bowled Samuels for no score in the same over.
Sharma was the first to strike when he had nightwatchman Fidel Edwards (one) caught behind before Yadav dismissed Kirk Edwards (33) and wicket-keeper Carlton Baugh (seven).
The tourists had gained a 95-run lead when they bowled India out for 209 in their first innings on Monday after posting 304.
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