The tourists got off to the perfect start to the series, thanks to three early wickets from Pat Cummins to have Bangladesh sitting at 3-10.
However, the hosts rallied, with a 155-run partnership from Tamim Iqbal (71) and Shakib Al Hasan (84) to even up the contest.
Lyon claimed his 250th Test wicket, overtaking Richie Benaud and becoming Australia’s second highest wicket-taking spinner, behind only Shane Warne.
With nine overs left to play in the day, Australia went out to bat and had the worst possible start.
Warner was dismissed LBW for just eight in the sixth over, before No.3 Usman Khawaja followed the very next over, run out for just one in calamitous circumstances.
Warner successfully overturned an incorrect LBW decision that he had cleared edged onto his pads, only to be out the very next ball.
In the next over, Khawaja did not offer a shot, before then bizarrely opting to take off for a run, only to be called back, before being caught short of his crease.
Nightwatchman Nathan Lyon was then also removed LBW for a duck just five balls later.
At stumps, Matt Renshaw is six not out, while captain Steve Smith is not out on one run.