Bangladesh went into the match having already achieved a first ODI series win against Pakistan, and made more history.
Opener Soumya Sarkar struck his maiden century as Bangladesh raced to 251-2 in the 40th over, after bowling out Pakistan for 250 in 49.
Soumya was not out on 127 off 110 balls, smashing 13 fours and six sixes.
Fellow opener Tamim Iqbal, who hit consecutive centuries in the last two matches, made 64 to take his run tally to 312, the highest for a Bangladeshi in any bilateral series, and the third highest in any three-match series. Tamim hit eight fours and one six in his 76-ball innings.
Tamim and Soumya shared 145 in their opening stand, to set the platform for Bangladesh's 10th clean sweep in its history.
"It is the best achievement in our history," Bangladesh captain Mashrafe Bin Mortaza said. "It was a perfect series for us in which everything went in the right direction. Everybody played their role perfectly.
"Now it's the time to show our credentials in T20s and tests."
However, Bangladesh effortlessly pushed on, as Mushfiqur Rahim joined Soumya to pile the misery on the listless Pakistani bowling attack. They shared an undefeated 97-run partnership, and Rahim fittingly hit the winning boundary off Junaid. Rahim was not out on 49 off 43 balls, including six fours.
"Tamim Iqbal told me to complete the match," said man of the match Soumya. "I tried to play my natural game and when I got in my zone, I didn't hesitate to play my shots."
Earlier, Pakistan captain Azhar Ali spearheaded his side's effort with 101, but Pakistan wasted a good start, and lost its last seven wickets for 47 runs.
Azhar's maiden century came off 111 balls, hitting 10 to the boundary. He shared in two significant partnerships before Bangladesh hit back.
Azhar and fellow opener Sami Aslam, on debut, combined for 91.
Aslam made a fluent 45, hitting seven fours off 50 balls. Both openers dealt with the Bangladeshi spinners with relative ease, until Nasir Hossain got the faintest of nicks off Aslam's thumb in the 18th over.
Arafat Sunny then continued Mohammad Hafeez's horror run of form, bowling him with a straight delivery when he was on 4.
Azhar, however, helped the side recover with the help of Haris Sohail. They shared 98 runs for the third wicket. Sohail struck 52 off 58 with one four and two sixes.
In the process, Azhar brought up his first ODI century with a double, flicking his counterpart Mashrafe Bin Mortaza towards mid-wicket. Azhar became the first Pakistan captain to hit an ODI century since Shahid Afridi in 2010.
Azhar's joy was short-lived as, on the next ball he faced, Shakib Al Hasan removed him with an arm-ball to leave Pakistan still well poised on 203-3 in the 39th over.
"We played well in patches but we could not put it together," Azhar said. "As everybody knows, the senior players have left. The youngsters have to stand up and we have to pull our heads together. We have some experienced guys for T20s and tests. We will put in a better performance."
Sohail departed four runs later, slicing Mashrafe for an easy catch.
The double dismissals sparked a sensational collapse. Pakistan lost its last six wickets for 43 runs on a pitch considered favorable to batsmen.
Left-arm spinners Shakib and Sunny, and fast bowlers Mashrafe and Rubel Hossain took two wickets each.