Resuming on 99 overnight, the 38-year-old reached his hundred with a single in the second over of the morning session on day two of the second Test, News.Az reports, citing foreign media.
Mushfiqur removed his helmet, raised both hands, and dropped to the ground in celebration.
His achievement places him among an exclusive group of cricketers who have scored a century in their 100th Test, joining legends such as Colin Cowdrey (104), Javed Miandad (145), Gordon Greenidge (149), Alec Stewart (105), Inzamam-ul-Haq (184), Ricky Ponting (120 & 143*), Graeme Smith (131), Hashim Amla (134), Joe Root (218), and David Warner (200).
It was the 13th Test ton for Mushfiqur, who is now level with Mominul Haque for the highest number of centuries in Test cricket for Bangladesh.
He eventually fell to left-arm spinner Matthew Humphreys for 106, edging to second slip where Andy Balbirnie completed a sharp catch.
The Ireland team offered warm applause as Mushfiqur left the field to loud cheers from the home crowd at the Sher-e-Bangla National Stadium.
Mushfiqur faced 214 balls and struck five fours during his milestone innings, which helped ease the early pressure after Bangladesh slipped to 95-3 on the opening day.
Bangladesh won the first Test by an innings and 47 runs.
