Bashar, who came to the crease after Hannan Sarkar was dismissed lbw by Pedro Collins with the first ball of the match, shared a solid second-wicket partnership with Javed Omar that took Bangladesh to 101 for one at lunch.
Omar fell for 32, the second victim for Collins, but Rajin Saleh helped Bashar, dropped on 27, add 50 for the third wicket.
After a rain interruption, Bashar reached his third Test century before he was caught by Dwayne Smith off fast bowler Jermaine Lawson for 113, including 15 fours.
Bangladesh, still seeking their first Test victory, progressed to 227 for three but lost four wickets for 23 runs late in the day.
Saleh was out for 26 to occasional spinner Ramnaresh Sarwan, Faisal Hossain fell to Collins for five and Mushfiqur Rahman and Khaled Mashud were also dismissed cheaply.
Mohammad Ashraful fought a lone battle to end the day unbeaten on 65, with Mohammad Rafique not out 17.
Collins finished with figures of three for 75 from 20 overs.
