Dhaka: Australia lost opener Matt Renshaw and Usman Khawaja in a disastrous opening to their run chase of 265 in the final session of day three in Dhaka.
Renshaw was the first to go, lbw to Mehedi Hasan for five, before Khawaja completed a forgettable match with the bat, sweeping to Shafiul Islam off Shakib Al Hasan to be caught for one.
Opener David Warner looked to be Australia's key man, racing to a half-century to bring the runs required to under 200, while captain Steve Smith narrowly survived a stumping appeal as he eased his way into his innings.
Bangladesh had been bowled out for 221 shortly after tea, with Nathan Lyon taking 6-82 to finish with nine for the match.
Lyon collected the final two wickets to fall 20 minutes into the final session, but a respectable batting effort by the Tigers - led by Tamim Iqbal's 78 and their captain Mushfiqur Rahim's 41 - means the tourists will require their second-highest successful run chase in Asia if they are to avoid what would be a humiliating Test defeat.
Earlier, Lyon picked up a pair of wickets in the first hour of the day, trapping nightwatchman Taijul Islam (four) plumb in front before finding the edge of Imrul Kayes' bat, who was caught at slip by David Warner and exited for two.
Tamim hit three boundaries from the bowling of Cummins and registered a second half-century of the match to continue his strong form, but Australia's concerns grew when Josh Hazlewood walked off the field with a side issue one ball into his second over of the morning.
It remains to be seen how serious the injury is, though a side strain would almost certainly spell the end of his involvement in this Test.
Tamim had made his way to 78, following on from a fine 71 in the first innings, when Pat Cummins bent his back and produced a snorter that looked on first inspection to have beaten the bat completely.
Several Australians detected a noise however, and Steve Smith referred the not out call, and was rewarded when snicko technology indicated ball had brushed glove.
Soon after, Shakib (five) attempted to loft Lyon over the off side, but instead skied the ball toward Cummins at cover, who took a good catch.
It was a third wicket of the innings for Lyon, who bowled superbly as his team was rocked by a side injury to Josh Hazlewood.
The Tigers' plucky skipper Mushfiqur looked in good touch almost from the moment he came to the middle, smashing Lyon delightfully over midwicket for a massive six before settling in for an extended stay in the middle, sharing a fighting 43-run partnership with Sabbir Rahman (22) which dominated much of the middle session.
That was until Mushfiqur was unluckily dismissed, run-out at the non-striker's end for 41 when the ball clipped Lyon's fingers and went onto the stumps.
The captain's wicket triggered a mini-collapse, with Nassir Hossain caught behind off Ashton Agar without scoring and Sabbir's cameo ending moments later when he was caught in close to give Lyon his fourth, before he wrapped up proceedings after tea with the quick wickets of Mehedi Hasan (26) and Sahfiul Islam (9).