Chasing a challenging 278 for the win, England put on a resilient show and survived a late fightback from the Tigers to seal a 2-1 series win with a four-wicket victory at the Zahur Ahmed Chowdhury Stadium on Wednesday.
Sam Billings and Ben Duckett struck gritty half-centuries to put England in a comfortable position in the chase. But quick strikes by pacer Shafiul Islam in successive overs and then by skipper Mashrafe Bin Mortaza kept the game alive for the hosts.
But Imrul Kayes dropped a sitter at the slips with England 21 runs away to stun the partisan home crowd into silence.
The English batsmen then held their nerves before Ben Stokes (47 not out) clubbed two sixes off Shafiul to finish things off with flare.
Upon seeing that the track offered pace, Mashrafe introduced Taskin Ahmed into the attack but the young pacer struggled with his length and gave away 12 runs in his first over which contained two great-looking drives into the ropes from Vince.
But Vince fell 32 in the next over, trapped in front, and Duckett walked in.
The 63-run opening stand was followed by mature strokeplay from Billings and Duckett. The duo drove, cut, swept in supreme fashion to add 64 more for the second wicket.
Mosaddek took out Billings for a topedge but Duckett added another 45 with Jonny Bairstow (15).
Duckett was initially cautious playing Shakib Al Hasan, who went through the batsman's gates in the second ODI, but soon read the bowler and cracked back-to-back boundaries to power England to 100 runs away from the finish line.
Stokes and skipper Jos Buttler (25) took the Lions within sight of the shore with a 48-run stand. But the Tigers skipper seemed determined to make things hard for them by bagging the scalps of his English counterpart and Moeen Ali (1) in a muted celebration.
Stokes, adjudged Man-of-the-Series, and Chris Woakes (27 not out) then took their side over the line.
The win made sure that after six touring parties, England are the first to taste victory in Bangladesh.
"It was a decent total. Obviously, the toss was the real factor. Their spinners got a lot of turn. Our spinners could not get the ball to turn because of the dew," he said
"We tried till the end but England were too good. They played really well. Congratulations to them, they have been really good.”
The duo scored 85 off 11.5 overs, the highest seventh-wicket partnership at the venue and against England, to set up a record chase.
Wily legspinner Adil Rashid was brilliant for England, bowling with guile and improvising his line by reading the batsmen well. He finished the series with 10 wickets.
Rashid, adjudged Man-of-the-Match, returned with a career-best 4-43 to better his haul of 4-49 in the first match and help England reduce the Tigers to 192 for six.
England had asked the hosts to bat first as rain relented after two days of continuous drizzle that had threatened to put the mouth-watering tie in doubt.
The duo were harsh on short balls, cracking boundaries on both sides of the wicket to help Bangladesh get 50 from the powerplay - the 50th run coming from a huge Kayes six over cover off another short ball.
They scored 80 and surpassed their own record of 63 for Bangladesh's highest opening partnership against England. But Kayes went back for 46 in a soft dismissal when he flicked straight to square leg off Stokes.
Man-of the-match Rashid's introduction in the next over hinted that Buttler wanted to dry up the runs but Tamim, on 45, was undone by a short ball outside off-stump, tamely slapping it straight to cover.
Mahmudullah swung at another short ball outside off from Rashid and fell for just six, which was a blast over midwicket in the previous ball.
With the England bowlers often wayward, Sabbir Rahman and Mushfiqur batted maturely and kept the scoreboard ticking with singles and doubles, punctuated with boundaries, to lift the run rate to more than 5 per over.
Ali finished 1-42 without giving away any run on the offside and was a bit fortunate to see the back of Shakib Al Hasan, who was stumped after the ball popped out of Buttler's gloves and hit the stumps.
Nasir Hossain's stay at the crease was short as Bangladesh wobbled in dealing with spin but Mushfiqur and Mossaddek made amends down the order.
Mushfiqur (67 off 62) smashed four boundaries, hammered a six over deep square leg and stayed unbeaten with Mosaddek (38 off 39) to power his team to the big total.