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Put in to bat, the visitors rode on a 98-run opening stand between Mendis, who smashed 70 off 42 balls, and Danushka Gunathilaka, who made 42, to post an imposing total.
For Mendis, who hit six fours and three sixes, it was his second successive fifty and a career-best T20 score.
Number three Thisara Perera chipped in with a quickfire 31. Upul Tharanga, with a 13-ball 25, and Dasun Shanaka, who smashed an unbeaten 30 off 11 balls, lifted the score past the 200-run mark.
Soumya Sarkar, Mustafizur Rahman, Abu Jayed and Mohammad Saifuddin claimed a wicket each for Bangladesh.
Sri Lanka lead the two-match series 1-0.