The Sri Lankans reached the target of 194 runs with 20 balls remaining at Mirpur Sher-e-Bangla National Stadium in Dhaka on Thursday.
The Lankan openers built a strong base with a 53-run partnership before Danushka Gunathilaka was out at 30 off 15.
It was the maiden wicket for debutant slow left-arm orthodox spinner Nazmul Islam who struck again to take out Upul Tharanga (4).
The other opener Kusal Mendis was declared Player of the Match for scoring the only fifty for the winners.
Dasun Shanaka and Thisara Perera took the islanders over the line with 42 and 39, respectively.
Nazmul was the most successful Bangladesh bowler with 2 for 25 in four overs.
All other Bangladesh bowlers proved expensive, giving away at least 10 runs per over, with Rubel Hossain bleeding 52 off 3.4 overs.
With four debutants in the team, the Tigers scored 193 for 5 in the first match of the series, bettering their previous best of 190-5 against Ireland in 2012.
Until Thursday, the highest total against Sri Lanka was in 2013, when they posted 181-7.
Soumya Sarkar hit his first T20I half-century while Mushfiqur Rahim was unbeaten on 66. Skipper Mahmud Ullah made 43. (Courtesy bdnews24)