Accompanying Mominul at the crease was Najmul Hossain Shanto, who scored his first Test century on the first day and added to it in the Thursday morning session to pass 150 at Kandy.
Bangladesh reached 474 for 4 before play was called off, with as many as 25 overs left in the day, due to bad light as regulations prevent the use of artificial light.
After lunch, the Tigers resumed at 378 for 2 and Sri Lanka found some success.
Bangladesh lost both Mominul and Shanto, but the tourists were well past 400 by then as the duo had added a record 242-run stand, the highest partnership for the third wicket for Bangladesh in Tests.
Lahiru Kumara ended the partnership by dismissing Shanto with the left-hander popping a simple return catch after being early into a drive.
Runs kept coming at a fair clip before Dhananjaya de Silva struck, finding Mominul's edge to slip.
At stumps, Mushfiqur Rahim was not out on 43 with Liton Das as his partner on 25.
Bangladesh won the toss on Wednesday and chose to bat. Despite the early dismissal of Saif Hassan, the Tigers managed a strong start through Tamim Iqbal and Shanto.
When Tamim was dismissed for 90, he was replaced by Mominul, who carried on where he left off.
Vishwa Fernando is the only Sri Lankan bowler to have found success on Day 1, picking up the wickets of Saif and Tamim for 75 runs.
Bangladesh will play two Test matches in Kandy during their tour of Sri Lanka. The second is scheduled to begin Apr 29.