At stumps on day one, Bangladesh reached 6-253, with Lyon finishing with figures of 5-77 from 28 overs, while Ashton Agar claimed the only other wicket.
Australia went into the match with three specialist spinners and just one recognised fast bowler in Pat Cummins. It proved immediate dividends with Lyon, who opened the bowling alongside Cummins, grabbing two early wickets to have the hosts sitting at 2-21.
Lyon in fact grabbed the first four wickets, but Soumya Sarkar (33), Mominul Haque (31) and Shakib Al Hasan (24) all made starts and had the score moving along on the turning deck.
Mushfiqur Rahim (62 not out) and Sabbir Rahman (66) then shared a 105-run partnership to put Bangladesh in a solid position, before Rahman was removed late in the day.
Lyon’s five-wicket haul is his fourth of 2017. If he can take one more, he will be the first Australian to take five five-wicket hauls in a year since Shane Warne in 2005.
The 29-year-old is now seventh on the all time list of Australian Test wicket takers, after overtaking former paceman Jason Gillespie.