The 25-year-old skewed the ball into the hands of Soumya Sarkar off the bowling of Taijul Islam, bringing wicketkeeper Matthew Wade to the crease to fight for the remaining 75 runs with Glenn Maxwell.
Set an imposing 265 to win on a deteriorating pitch, Warner scored an unbeaten 75 to shift the momentum of the match and lift the visitors to 2-109 at stumps on day three.
Skipper Steve Smith, who survived a tight stumping decision from his first ball, reached 25 not out, leaving Australia needing a further 156 runs to pull off an unlikely triumph.
Both Warner and Smith were given a life by a Bangladeshi side that earlier in the day had been deserved favourites to beat Australia in a Test match for the first time.
Khawaja (one) continued his disastrous return to national duties, lofting an ill-judged sweep shot to Taijul Islam at deep backward square leg off Shakib’s left-arm spin.
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