Lyon, who had taken five wickets on the opening day, returned impressive figures of 7-94 including the prized scalp of skipper and overnight batsman Mushfiqur Rahim for 68.
The hosts, who lead the series 1-0, added 52 runs to their overnight score of 253-6 before losing all their 10 wickets in the first session of play.
In reply, Australia were 15-1 at lunch with David Warner on 2 and skipper Steve Smith on 8 at the crease. The visitors still trail Bangladesh by 290 runs in the first innings.
Earlier Lyon struck on his second delivery of the day to get Rahim bowled on a tossed up delivery that spun and got the inside edge of the batsman's bat before whipping the bails off.
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Wicketkeeper-batsman Rahim added just six runs to his overnight score of 62.
Rahim's overnight partner Nasir Hossain, with whom he shared a 43-run seventh-wicket stand, made a 97-ball 45 before becoming left-arm spinner Ashton Agar's second victim.