The tourists, trailing by 488 runs on the first innings, were 110-1 in their second knock at lunch on the third day at the Sinhalese Sports Club here on Wednesday.
Omar has so far hit nine boundaries in his eighth Test half-century. Rajin Saleh was the other batsman at the crease on 12.
Bangladesh still need 378 runs to make Sri Lanka bat a second time, but they showed a willingness to put up a fight after being shot out for 89 in their first innings.
Openers Omar and Shahriar Nafees, who resumed at the overnight score of three for no loss, frustrated the Sri Lankans by carving out 86 for the first wicket.
It was Bangladesh's best opening stand against Sri Lanka, surpassing the 52 between Omar and Nafees Iqbal at home in Bogra last year.
Omar began by driving Chaminda Vaas through the covers thrice in one over and then flicked two consecutive boundaries off Lasith Malinga at the other end.
Left-handed Nafees straight drove Vaas and pulled Malinga before bringing up the 50 partnership in 55 minutes by lofting off-spinner Muttiah Muralitharan for a four.
Muralitharan, who bagged five wickets in the first innings, finally broke through 30 minutes before lunch when he accepted a hard return catch off a Nafees straight drive.
Nafees made 38 with seven boundaries.
Sri Lanka had piled up 577-6 declared in their first innings with captain Mahela Jayawardene, opener Michael Vandort, wicket-keeper Prasanna Jayawardene and Vaas scoring centuries.
Bangladesh and Sri Lanka will play three Tests and as many one-day internationals on the five-week tour.