Tendulkar was bowled by West Indies spinner Sunil Narine in the fifth over as Mumbai chased the hosts' 159-6 at Eden Gardens.
His opening partner Dwayne Smith smashed five sixes in a swashbuckling 62 off 45 balls but was dismissed at the end of the 11th over with the score on 82.
His fellow West Indian Kieron Pollard made 33 to leave the visitors needing 10 off the final over but he was dismissed by Rajat Bhatia's opening delivery of the final stanza.
However Harbhajan Singh swiped his second ball for six over mid-wicket before Ambati Rayudu scored the winning runs off the penultimate ball.
Earlier the Knight Riders had steady contributions from their middle order of Jacques Kallis (37), Manoj Tiwary (33) and Eoin Morgan (31) in setting a competitive target.
Mumbai now have four wins and three losses while Kolkata slip to their fifth defeat and stand well outside the four play-off places, albeit less than halfway through the 16-match group stage.
- Sports & Recreation
- Dwayne Smith
- Sunil Narine
- Rajat Bhatia
- Harbhajan Singh