The hosts, chasing an imposing first-innings total of 590 by West Indies, were 281 for three at the close of the third day with Tendulkar (67) and VVS Laxman (32) at the crease.
The 38-year-old Tendulkar, with 51 centuries in Tests and 48 in one-dayers, needs one more to get to the coveted ton that has eluded him since he last made three figures in the 50-over World Cup in April.
Playing in front of a 20,000 crowd at his home Wankhede Stadium ground, the master batsman hit five boundaries and a six as he added 57 for the fourth wicket with Laxman.
Tendulkar's only blemish came when he edged leg-spinner Devendra Bishoo on 58 but wicketkeeper Carlton Baugh failed to hold on to the tough chance.
Dravid (82) became the second player after Tendulkar to reach 13,000 runs in Tests, and also brought up 1,000 runs in a year for the third time in his career as he made his 62nd half-century.
He added 86 runs for the third wicket with Tendulkar before being clean bowled while attempting to cut part-time spinner Marlon Samuels.
The 38-year old Dravid also put on 71 for the second wicket with Gautam Gambhir (55) after India lost Virender Sehwag (37) before the lunch break.
Dravid went past the 13,000-mark with a sublime cover drive off Darren Sammy, one of his 11 boundaries.
Gambhir hit eight boundaries before he edged to Baugh after trying to hook a high bouncer by paceman Ravi Rampaul.
Sehwag began in typically explosive fashion, striking three fours and a six before he was bowled by West Indies captain Sammy.
Medium-pacer Sammy, who was swatted for a six by Sehwag in his first over, managed to breach the batsman's defense with a ball that moved back off the pitch after the openers had put on 67.
Sehwag and Gambhir took India past 50 in 11 overs but the hosts slowed considerably after the fall of the first wicket, adding seven runs in 4.3 overs up to the break.
Earlier, India off-spinner Ravichandran Ashwin completed his second five-wicket haul in three Tests as West Indies were finally bowled out in the morning.
Resuming on 575 for nine, the visitors added 15 runs in 3.1 overs before Bishoo was bowled for 12 by Ashwin.
Ashwin, 25, who grabbed a maiden five-wicket haul on his debut in the first Test in Delhi, finished with five for 156 from 52.1 overs.
Building on half-centuries from the top six batsmen, including Darren Bravo's classy 166, West Indies have almost certainly ended any hope India had of inflicting a 3-0 series whitewash.
