They eventually succeeded in their mission as Mominul followed up his 176 in the first innings with 105 and Liton made 94 after Bangladesh had resumed on 81 for three.
The duo batted through the morning session to frustrate Sri Lanka, their 180-run stand coming to an end when spinner Dhananjaya de Silva had Mominul caught in the slip.
Stand-in skipper Mahmudullah faced 65 balls in his unbeaten 28 as Bangladesh reached 307-5 when players from both sides shook hands with 17 overs remaining in the batsman-dominated contest in which 1533 runs were scored for 24 wickets in five days.
Dhaka hosts the second and final test from Thursday.
(Reporting by Amlan Chakraborty in New Delhi; editing by Toby Davis)
