Mehidy returned with 4-12, his best figure in T20I while Shakib al Hasan, Mustafizur Rahman, Taskin Ahmed and Hasan Mahmud got one wicket apiece.
Mehidy, who got some assist from the slow and low pitch of SBNS, maintained ideal length with a bit of turn that proved extremely difficult for the English batters.
Left-hander Moeen Ali was his first victim when the batter tried to slog and got caught at deep by substitute Shamim Patwary for 15 and leaving his side on 57-4 after the end of ninth over.
Mehidy must thank wicket-keeper Liton Das for his next two wickets as the latter showed first glove work to the extra bounce that beat Sam Curran to complete the stumping and also got rid of Chris Woakes two balls later in similar manner.
Mehidy got his last wicket in the final ball of his spell when Chris Jordan was caught at deep by Rony Talukdar.
Ben Duckett, who held one side of the wicket for long, was the highest scorer for England with a run-a-ball 28 while opener Philip Salt scored 25 off 19.