Fast bowler Mluleki Nkala led a brief Zimbabwean recovery with two wickets in the afternoon session, but Bangladesh's skipper Bashar soon restored the balance with his impressive knock in a clash between the Test minnows.
Bashar looked set to complete his fourth Test hundred when he edged debutant seamer Christopher Mpofu to be caught by wicket-keeper Tatenda Taibu, who dived to his right to bring off a low catch.
Bashar made the most of an inexperienced Zimbabwean attack as he hit 14 fours in his 128-ball knock, while Saleh struck one six and seven fours in his third Test half-century.
The hosts got off to a solid start after winning the toss on a slow pitch as Nafis Iqbal (56) and Javed Omar (33) put on 91.
Their stand was Bangladesh's best for the opening wicket in 35 Tests, surpassing the previous highest of 73 between Al Sahariar and Omar against the same opponents at the same venue in 2001.
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