The hosts surpassed the seemingly improbable victory target of 387 on a wearing wicket with 20.3 overs to spare on the fifth and final day to take a 1-0 lead in the two-match series.
Tendulkar swept off-spinner Graeme Swann to fine-leg for the winning boundary that also brought up his 41st Test century amid loud cheers from 30,000 home fans at the Chidambaram stadium.
Tendulkar, showing the form that has made him the all-time leading scorer in both Tests and One-day cricket, anchored India's chase for five hours, during which he hit nine boundaries.