Dawid Malan and Jos Buttler’s 95-run stand carried England to 100 for one after 13 overs in pursuit of 159 but the pair were dismissed off back-to-back balls before the wheels fell off in this dead rubber.
The tourists lost five for 28 as their challenge faded even if, needing 27 from the final over, Chris Woakes hit the first two balls for four before they subsided to 142 for six.
England’s threadbare batting has been a constant feature in this series and will be the key talking point again after failing to build on Malan’s 53 off 47 balls and 40 off 31 deliveries from Buttler, batting at three for the first time in his international career as part of a rejigged top order.
Buttler had earlier won the toss for the first time in nine attempts this year but an unchanged England delivered a slipshod fielding performance replete with basic errors on the boundary while Rehan Ahmed and Ben Duckett spilled simple catches, both off Jofra Archer.
Litton Das, dropped by Duckett on 51, top-scored with 73 off 57 balls, putting on 84 with Najmul Hossain Shanto, who contributed an unbeaten 47 from 36 deliveries in Bangladesh’s 158 for two.
It proved to be a winning score despite England’s seamers finishing strongly in their final assignment of a long winter, allowing Bangladesh just 27 in the last five overs.
Litton settled after a streaky edge brought the first of 10 fours, with Buttler turning to Adil Rashid in the third over following ineffective opening bursts from Sam Curran and Woakes.
Archer should have been in the wickets column but was left covering his eyes when Ahmed settled under a skier only for the ball to drift through his fingertips to spare Rony Talukdar, who was on his way for 24 after a reverse sweep off Rashid caught a toe-end and ballooned back to the bowler.
Najmul looked to inject some urgency by slog-sweeping Moeen Ali and then hammering fellow spinner Ahmed for sixes either side of Litton going to a 41-ball fifty. Litton had a reprieve shortly after as a charging Duckett could not hold on in the deep, bringing a wry smile out of Archer.
Litton soon had more good fortune with an edge over Buttler before middling one into the stands, but the opener’s luck expired when he pulled a Chris Jordan slower ball into the pitch to Phil Salt.
England’s misfields meant Archer’s figures of 4-0-33-0 was particularly unfortunate but he played his part to shackle Najmul and Shakib Al Hasan at the death.
Yorkers from Jordan and Curran, who conceded just four singles in an excellent final over, were also supremely effective.
Salt was stumped for a golden duck having resumed his opening partnership with Malan, who was given lbw after Taskin Ahmed thudded into his pads. Malan hesitantly reviewed and the on-field decision was overturned despite uncertainty about whether a spike was detected after the ball passed the bat.
Bangladesh did not seem overly perturbed and Malan capitalised with a slog-sweep off Bangladesh debutant Tanvir Islam for six. Buttler timed Taskin and Shakib to the boundary as he and Malan settled into a rhythm.
With Mustafizur Rahman proving difficult to get away, Buttler and Malan each dispatched Taskin for fours before the England captain cleared the rope with a straight hit off Mehidy Hasan Miraz.
Malan brought up a 43-ball fifty but both set batters perished in successive deliveries.
Malan feathered a pull through to Litton off Mustafizur before Buttler was short of his crease after Mehidy’s direct hit, finding the one stump he had to aim at despite throwing off-balance from point.
It proved the turning point in the contest.
While Moeen and Duckett initially restored some order with a couple of boundaries, they were both dismissed in the same Taskin over. Moeen holed out to deep midwicket while Duckett lost his off-stump when Taskin went full and straight.
Woakes dispatched Hasan Mahmud for consecutive fours at the start of the 20th over to leave England needing 19 off four balls but the Bangladesh quick recovered his composure to help his side get over the line.