Mortaza hit 44 off 29 balls after Mahmudullah Riyad top-scored with 75 off 88 balls, leading Bangladesh to 238-8 at the Sher-e-Bangla National Stadium.
Mortaza then brought his bowling skill into play, picking up 4-29, including three wickets in his opening spell that left England tottering at 26-3.
England couldn't recover from the position before they were all out for 204 runs in 44.4 overs, despite the best efforts by captain Jos Buttler, who scored the highest 57 runs for the visiting side.
Taskin had earlier removed Jonny Bairstow for 35 to end his 79-run fifth-wicket partnership with Buttler as the duo attempted to revive England's innings after the initial damage.
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Buttler hit Taskin for three of his seven fours in one over before the bowler came back strongly to finish with 3-47.
"I think we were a bit tentative about it (the run chase) as a side," said a disappointed Buttler after the match.
"We wanted to come out on the front foot and take it on. We thought that was chaseable. We bowled well up front but it was a good wicket and 240 was attainable."
"It wasn't much of a total but we could defend," said Mortaza, who also won the man of the match award.
"If you look at the other match, to come back was really professional," he said, referring to the first match when Bangladesh came close to chase down England's 308-9 before losing their way.
Earlier Mahmudullah led Bangladesh's stop-start batting when he came to bat at number four after Chris Woakes struck twice in his opening spell.
Ball, man of the match on Friday with five wickets on debut, kept up the pressure when he bowled Sabbir Rahman for three.
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