Bangladesh's lead over Sri Lanka nears 200 in first Test
The visitors finished at 177-3 after a strong partnership of 68 between opener Shadman Islam and skipper Najmul Hossain Shanto.
All-rounder Milan Rathnayake dismissed Shadman for 76 but veteran campaigner Mushfiqur Rahim, fresh off a first-innings century, joined his captain to steady the innings.
The duo built an unbroken 49-run partnership for the fourth wicket to put their side in the driver's seat heading into the final day.
Bangladesh off-spinner Nayeem Hasan earlier took five wickets to halt Sri Lanka's first-innings charge, with the visitors dismissed for 485 after lunch to trail by 10 runs.
Nayeem was well supported by Hasan Mahmud, who finished with 3-74.
Sri Lanka looked set for a useful lead after the first session, reaching 465-6 with Kamindu Mendis and Rathnayake stitching together a gritty 84-run stand for the seventh wicket.
However, Sri Lanka then surrendered the momentum with a lower-order collapse, losing their last four wickets for 15 runs.
Rathnayake had shown commendable defiance for his 39 but was undone by an inside edge onto the stumps off Mahmud.
Kamindu followed in the next over with Sri Lanka's score unchanged, falling to Nayeem's off-spin as the ball kicked off the surface and caught the left-hander by surprise.
His composed 87 included eight fours and a six but he fell with a third hundred against Bangladesh within sight.
Nayeem said he "really enjoyed" Kamindu's wicket.
"Very happy with that ball that got him," he said.
Nayeem also removed debutant Tharindu Rathnayake for a duck with another sharp off-break that went through the gate as Sri Lanka lost three wickets for just one run in 10 deliveries.
He then bowled Asitha Fernando (4) to close out Sri Lanka's innings.
The last Test draw at the Galle cricket ground was between the same two sides in 2013.
"I think the game is open for both sides," Kamindu said.
"If we are able to get three wickets early, then we are up for a challenge."
Brief scores:
Bangladesh 1st innings: 495 all out (Mushfiqur Rahim 163, Najmul Hossain Shanto 148; Asitha Fernando 4-86)
Sri Lanka 1st innings: 485 all out (Pathum Nissanka 187, Kamindu Mendis 87, Dinesh Chandimal 54; Nayeem Hasan 5-121, Hasan Mahmud 3-74)
Bangladesh 2nd innings: 177-3 (Shadman Islam 76, Najmul Hossain Shanto 56)
Toss: Bangladesh