Lyon removed Bangladesh's top order, including opener Tamim Iqbal trapped in front, as Australia pegged back the hosts in the first session.
Australia suffered their first-ever Test defeat against Bangladesh last week and must win in Chittagong to draw the series.
Bangladesh opener Soumya Sarkar reached 33 before he was out on the stroke of lunch. Mominul Haque was the not out batsman on 24.
The pair had added 49 in 16 overs before Soumya was plumb in front to a skidding delivery by Lyon.
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He never looked comfortable at his home ground, and was dropped at third slip by Glenn Maxwell off fast bowler Pat Cummins.
Imrul Kayes made just four before he was sent back to the pavilion after Australian skipper Steve Smith reviewed a not-out decision by English umpire Nigel Llong.
Bangladesh captain Mushfiqur Rahim won the toss and opted to bat first.
Bangladesh brought in Mominul in place of pace bowler Shafiul Islam.
Australia, trying to save the series, deployed a three-pronged spin attack with Steve O'Keefe replacing injured fast bowler Josh Hazlewood.