Sri Lanka, replying to Bangladesh's first-innings total of 232, were 5-375 at stumps on Tuesday's second day of the first Test at the Sher-e-Bangla stadium in Dhaka.
Kaushal Silva hit 139, his maiden Test century, Kumar Sangakkara made 75 and Dimuth Karunaratne scored 53.
Sri Lanka ended the day with a lead of 143 runs with five wickets in hand.
Silva led a dominant batting display, the right-hander putting on 118 for the first wicket with Karunaratne and then teamed up with veteran Sangakkara to add a further 155 runs for the second wicket.
Sri Lanka will resume on Wednesday looking to grab a sizeable lead and have the hosts on the ropes in the first match of the two-Test series.
Bangladesh's bowlers, who managed just one wicket in the morning session on the even-paced pitch, claimed four in the post-tea period to slow Sri Lanka's run spree.
Left-arm spinner Shakib Al Hasan was the pick of the bowlers with 3-94, including the vital wicket of Silva.
Sri Lanka, who started the day on 0-60, had moved to 118 when left-hander Karunaratne was snapped up at short cover off Shakib just before lunch.
After a barren post-lunch period, Bangladesh removed Silva and Sangakkara in quick succession in the final session. They claimed two more wickets as Sri Lanka slipped from a comfortable 1-273 to 5-375.
Mahela Jayawardene was unbeaten on 42 when play was called off soon after the dismissal of nightwatchman Suranga Lakmal in the final over from Sohag Gazi.
The 27-year-old Silva reached the three-figure mark in his seventh Test after making 95 and 58 in the Dubai Test against Pakistan earlier this month.
But he was lucky to achieve the landmark after being dropped by Bangladesh captain Mushfiqur Rahim behind the stumps and again by Shamsur Rahman early in his innings.
Silva had another reprieve before lunch when he was caught behind off a no-ball from seamer Al-Amin Hossain.
He moved from 85 to 97 with a flurry of boundaries and then reached his century by pulling Shakib to the fine-leg fence for another four.
Sangakkara looked set for his 34th Test hundred until he edged a catch to the slips off Al-Amin, who was finally rewarded for his hard work in unresponsive conditions.
Silva, who hit 16 boundaries and two sixes, was dismissed five overs later when he was trapped lbw by Shakib.
Dinesh Chandimal helped Jayawardene add 72 for the fourth wicket before being bowled by Shakib for 40.
The two Test matches will be followed by two Twenty20 matches and three one-day internationals.
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