Chasing 276 for victory, VVS Laxman (58 not out) and skipper Mahendra Singh Dhoni (0) were at the crease when India successfully completed the chase to take a 1-0 lead in the three-match series, on the fourth day of the match.
Off-spinner R Ashwin, who took match haul of nine for 128, including six in the second innings, also played a big role in India's triumph as he became only the third player to win the man-of-the-match award on debut.
Resuming at 152 for two and needing another 124 runs to win with eight wickets in hand, India lost Rahul Dravid (31) early on the fourth day.
But Tendulkar along with Laxman guided the hosts to 233 with a 71-run partnership before disaster struck some time before the lunch break.
India were just 43 runs away from achieving a hard-fought come-from-behind victory and take a 1-0 lead, when Tendulkar made his way back to the dressing room, much to the disappointment of the crowd.
But Laxman stood firm to guide the Indians with yet another half-century in the fourth innings of a match. He struck six boundaries and faced 105 balls.
Ashwin, who was declared man of the match on his Test debut for his nine wickets, said, 'I expected to get a few wickets but not so many. It is a very happy feeling to go into my wedding with such a performance, the only thing was the variation in bounce, if you were slower through the air, you would be slower off the pitch. So I bowled a bit quicker in the second innings. It helped that Ojha was bowling so many maidens from the other end. The batsman did not have the fear of being caught at bat-pad.'
An ecstatic Indian skipper Dhoni, said, 'That was a big lead, We all felt we did not bat the way we should in the first innings, we said we will have to make it up. Ojha bowled well in the first innings, Ashwin has got a few variations and he was supported by the other bowlers. We got a good start in both innings, and the middle order did well in the second. It was not an easy wicket to score, you had to be aware of the ball that kept low. We have got one more day of rest, we will relax and get ready for the next one.'
On his part, Windies captain, Darren Sammy, said, 'When you play against a top team, you want to create the opportunity to win, we did that in the first innings but did not bat well in the second. The pitch played much better on the third and fourth days. We want improvement, we have to look to create opportunities and hopefully cash in. We believe we can win matches here.'