Dasun Shanaka and Thisara Perera gave Bangladesh's bowlers a torrid time to build on Kusal Mendis' half-century as Sri Lanka achieved their biggest Twenty20 international run chase to earn a six-wicket win.
Kusal's 53 off 27 set the tone in Sri Lanka's pursuit of 194, before Shanaka (42 not) and Thisara (39no) finished the job in Mirpur to help the visitors to a 1-0 lead in the two-match series.
Earlier, Mushfiqur Rahim's 66 and contributions of 51 and 43 from Soumya Sarkar and Mahmudullah respectively had helped Bangladesh to post a seemingly imposing total of 193-5 from their 20 overs.
Sarkar put on stands of 49 and 51 with Zakir Hasan and Mushfiqur before an ugly reverse sweep helped Jeevan Mendis strike lbw with his first ball and left the batsman needing help from team-mates to leave the pitch.
Mushfiqur took the attack to Sri Lanka and smashed seven fours and a six from his 44-ball knock, ably assisted by Mahmudullah as part of a fruitful 73-run stand, before the latter scooped Akila Dananjaya to short fine leg.
Kusal - playing in place of the injured Kusal Perera - wasted little time in making inroads into Bangladesh's total, striking eight fours and a couple of sixes before having his progress halted when he holed out to long-off from the bowling of debutant Afif Hossain.
That sparked a mini-blip as Sri Lanka lost three wickets for 39 runs, but Shanaka made 42 off 24, and Thisara 39 off just 18 as part of a match-winning, unbeaten stand of 65.