Mushfiqur hit 106 and added an unbeaten 53 in the second innings for a match total of 159, while Liton Das struck 128 as the hosts posted 476 in their first dig.
Irish offspinner Andy McBrine took 6-109 to stall Bangladesh's push past 500, but Taijul Islam dominated the match with figures of 4-76 and 4-94, becoming the first Bangladeshi bowler to reach 250 Test wickets as Ireland were dismissed for 265 in their first innings.
Bangladesh, however, opted to not enforce the follow-on and openers Shadman Islam (78) and Mahmudul Hasan Joy (60) combined for a 119-run partnership.
Mominul Haque then led the charge with 87, his third consecutive 50-plus knock in the series, having scored 87 and 63 in the last two innings, as Bangladesh declared at 4-297, setting Ireland an improbable 509 for victory.
Ireland pair Harry Tector (50) and Curtis Campher dug in to take the game into the fifth day.
Campher, who was left stranded on 71, shared half-century partnerships with Jordan Neill (30) and Gavin Hoye (37) to delay Bangladesh's push for victory on Sunday.
Hasan Murad took 4-44, including wickets in successive deliveries to wrap up the Irish second innings for 291.
Bangladesh won the first match by an innings and 47 runs.
The two sides play a three-match T20 series next, beginning on November 29 in Chattogram.
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