The loss sent India crashing out of the tournament. Sri Lanka must wait on the result of the last Super Eight match later Tuesday between Australia and the West Indies to discover its fate.
Chamara Kapugedera struck an unbeaten 37 off 16 balls, including a six off the final delivery to inspire Sri Lanka to its winning total of 167-5 off 20 overs after India had rattled up 163-5 in its innings.
Angelo Mathews hit an equally crucial 46 off 37 deliveries while skipper Kumar Sangakkara anchored the chase with 46 off 33 balls.
Paceman Vinay Kumar led India's bowling figures with 2-30 off four overs, while left-hander Suresh Raina topscored with 63 off 47 balls.
India, needing to win by 20 or more runs to keep its chances of a final four spot alive, had made a spirited start to defend its total as Sri Lanka slumped to 6-2 after two overs.
Ashish Nehra removed the in-form Mahela Jayawardene (4) to an edge to first slip and Kumar then added the scalp of Sanath Jayasuriya (0), who pulled to midwicket.
Sangakkara led the recovery in a stand of 43 with Tillakaratne Dilshan (33) and a further association of 56 with Mathews.
Sangakkara's dismissal, at 105-4 in the 15th over, left the match in the balance.
Mathews and Kapugedera first saw Sri Lanka past the score of 144 that eliminated India and then enjoyed a series of sixes to seal the victory.
Their stand was worth 56 before Mathews was run out off the penultimate delivery.
Sri Lanka needed three to win off the last ball and Kapugedera hit a huge six over extra cover off Nehra.
