Sri Lanka could not come out to bat for the second time in the match as almost the entire final session was lost to rain.
Resuming day three on 133-2, Bangladesh lost four wickets in the morning session but captain Mushfiqur waged a grim battle against the Sri Lankans.
The stumper-batsman joined forces with Mehedi Hasan Shiraz (41) to raise 106 runs for the seventh wicket, only the second 100-plus partnership of the innings, to help them cross the 300-mark.
Mushfiqur brought up his 16th test 50 off 107 balls but fell to rival skipper Rangana Herath who claimed 3-72.
Off-spinner Dilruwan Perera also claimed three wickets, dismissing Mehedi and Taskin Ahmed in successive deliveries.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Alison Williams)
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