Mendis smashed 53 off 27 balls while Dasun Shanaka hit an unbeaten 42 as the visitors eased past Bangladesh's record total of 193-5 with 20 balls to spare.
Shanaka, who hit 3 fours and 3 sixes in his 24-ball blitz, shared an unbeaten 65-run stand with Thisara Perera to snap Sri Lankas eight-match losing streak in T20 matches.
Bangladesh rode on brisk half-centuries by Soumya Sarkar and Mushfiqur Rahim to post their highest-ever T20 total but were undone by an attacking opposition batting.
Left-arm spinner Nazmul Islam, one of the four debutants in Bangladesh line up, broke the stand when he got Gunathilaka stumped.
Nazmul later picked the wicket of Upul Tharanga, for four, before fellow debutant Afif Hossain removed Mendis with his off spin.
But Shanaka and Perera, who was unbeaten on 39 off just 18 balls, ensured no further hiccup in the Sri Lankan innings.
Mushfiqur remained unbeaten on 66 off 44 balls while Soumya hit a 32-ball 51 -- the left-handed openers maiden T20 fifty -- after the hosts elected to bat first.
Sri Lankan leg-spinner Jeevan Mendis claimed two wickets.
The second match of the series will be held in Sylhet on Sunday.
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