Set a challenging 332 to win, England, who without most of their first-choice players because of a pre-series Covid outbreak within their camp, were faltering at 165-5.
Bangladesh, who lead the three-match series 1-0, were in trouble at 74-4 when Mushfiqur put on 87 runs for the fifth wicket with Mahmdullah Riyad (41), before they were bowled out in 48.1 overs in Dhaka.
Babar Azam became the fastest cricket player to reach the milestone of 2,000 runs in the Twenty20 international format, having accomplished the feat in just 52 innings.
A century stand between Mitchell Marsh and Glenn Maxwell and four wickets for spinner Adam Zampa helped Australia down world champions England by 19 runs.