Renshaw survived nearly two sessions, facing 94 balls for his 45 runs while Australia lost five wickets, until he edged to Soumya Sarkar at slip.
Australia was comfortably placed at 158-2 thanks to Warner's first century on the subcontinent, resuming the day on 109-2 and needing 156 runs on the fourth day.
But Bangladesh were back once that stand was broken.
"But you've got to keep trying to learn and keep trying to get better in these conditions". "They had aggression but we also showed that Bangladesh can be an aggressive side", he said. When they last toured Bangladesh, in 2006, they scraped home by just three wickets in the first Test in Fatullah.
David Beakley, the Australia physiotherapist, said of Hazlewood's injury, "This type of injury will require several weeks of rehabilitation". David Warner started the day at 75 not out alongside captain Steve Smith.
Australia's spin duo of Lyon (3-79) and Ashton Agar (3-46) shared six wickets between them to bowl out the hosts for a modest total.
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Shakib then took centre stage as he lured Glenn Maxwell (23) down the track before turning the ball away, Mushfiqur Rahim having the simple task of completing the stumping. The loss again highlighted Australia's historic batting issues on pitches in subcontinental Asia - a 2-1 series loss in India this year came after a 3-0 defeat in Sri Lanka in 2016.
Shakib is now the No. 1 all-rounder in the ICC Test rankings and he showed exactly why as he spun a web around the Australians after a fine 84 in Bangladesh's first innings.
Tailender Pat Cummins fought valiantly, belting offspinner Mehedi Hasan for two sixes in one over on the way to an unbeaten 33, but it was all over when Taijul Islam trapped Josh Hazlewood lbw for a duck. Shakib Al Hasan was the architect of Bangladesh's win as they clinched the series opener by 20 runs. "We got eight wickets, the fielders too chipped in".
SHAKIB Al Hasan was the hero as Bangladesh spun their way to a first Test victory over Australia.
Smith said failing to pile up runs in the first innings had made their job tougher.
Interestingly, this was also one of the most thrilling Test matches in recent times irrespective of the teams participating in them.