On a tough day for the Irish bowlers, pacer Graham Hume was the only standout performer, finishing with 4-60 in his 10 overs.
Sent to bat, Bangladesh lost captain Tamim Iqbal (three) early on, when he edged a Mark Adair delivery straight to Paul Sterling at slips.
Liton Das (26 off 31 balls) was looking good, before giving a regulation catch to Sterling off Curtis Campher and Najmul Hossain Shanto (25 off 34 balls) also threw away a decent start, getting clean bowled by Curtis Campher.
Bangladesh were in a spot of bother at 81-3 after 16.3 overs when Shakib and Towhid got together.
Shakib looked a bit circumspect, struggling to hit fours whereas Towhid looked in control from the get go.
The 22-year-old Towhid showed maturity beyond his years, kept rotating the strike and also struck boundaries in between.
Shakib, on the other hand, completed 7000 runs in ODI cricket in the 20th over and took 65 balls to complete his 53rd half-century in ODIs.
However, in just one over the southpaw made up for his slightly slow start. Campher came in to bowl the 35th over and Shakib took him to the cleaners, hitting him for five fours in an over that fetched Bangladesh and Shakib 22 runs.
But Shakib’s penchant to keep hitting boundaries brought his demise. The left-hander edged an overpitched ball from Hume to the wicketkeeper in the second ball of the 38th over to fall seven runs short of what would’ve been his 10th ODI century.
The wicket also ended a 135-run third wicket stand between Shakib and Towhid.
Towhid, who reached his fifty off 55 balls, and Mushfiq then added 80 runs for the sixth wicket off just 49 balls which ensured Bangladesh would post a daunting total.
Mushfiq and Towhid departed in space of three balls from Hume, the former getting caught and the latter losing his stumps to a yorker, as Bangladesh were reduced to 297-6 in 45.5 overs.
Yasir Ali then added 17 runs in the final overs to help Bangladesh post the record total.