Put into bat, Bangladesh suffered from a lack of partnerships but Sabbir's 50-ball 77 gave the team a competitive total at Colombo's R. Premadasa Stadium.
Leg-spinner Yuzvendra Chahal claimed three wickets to rattle the Bangladesh batting early including the prized scalp of the in-form Mushfiqur Rahim for nine. He returned figures of 3-18.
Sabbir then went after the Indian pacemen Vijay Shankar and Shardul Thakur, who gave away 93 runs between them.
Left-arm quick Jaydev Unadkat struck late to get two wickets in an over as he finally got Sabbir, who hit seven fours and four sixes, bowled on a slower delivery.
Bangladesh have never won against India in their seven T20 meetings so far.