Sent into bat first, Sri Lanka managed to accumulate 280 runs in the first innings despite a marathon 202-run partnership between Mendis and de Silva.
The touring side, after getting off to a shaky start, lost wickets at an alarming rate and thus consequently slipped to 57/5.
Following the debacle, Mendis joined his captain in the middle, and the duo, partnered strongly to pull the visitors out of danger.
They batted sensibly for more than 40 overs and recorded gutsy centuries in the process before Nahid Rana provided a much-needed breakthrough to Bangladesh by getting rid of Mendis.
Sri Lanka captain de Silva followed in his footsteps a couple of overs later as Bangladesh forced a comeback.
Kamindu Mendis and Dhananjaya de Silva remained the joint top scorers for Sri Lanka, scoring 102 runs each.
Bangladesh bowlers then ran through the toothless Sri Lanka batting tail and consequently booked the touring side on 280.
Khaled Ahmed and Nahid Rana led the bowling attack for Bangladesh with three wickets each while Tajiul Islam and Shoriful Islam struck out a batter apiece.
In response to 280, Bangladesh was in a state of bother when the stumps were called as the home side were 32/3 with opener Mahmudul Hasan Joy (9) and night watchman Taijul Islam (0) on the crease.