Right-arm seamer Kasun Rajitha took the wickets of Litton Das and Mosaddek Hossain in the same over to grab a five-for near the start of the day.
But Mushfiq fought on alongside Taijul Islam. At the end of 102 overs, the Tigers were 337/7. Mushfiq was on 157*.
The hosts were 24/5 inside the first hour on Day 1 before Litton Das and Mushfiqur Rahim staged an epic comeback and put Bangladesh on course for an imposing first-innings total.
The duo fashioned a 253-run sixth-wicket stand as Bangladesh reached 277 for five at stumps after choosing to bat at the Sher-e-Bangla National Cricket Stadium in Mirpur.
Litton (141) lost his wicket early on Tuesday.
The teams are playing two Test matches. The series forms part of the 2021–2023 ICC World Test Championship.
