Usman Khawaja rode his luck to an unbeaten 47, while Travis Head was on six at the close with Australia still 114 behind in what promises to be a low-scoring conTest.
The tourists could have been in a better position but for two avoidable dismissals in the final session.
On a pitch where spinners dictated terms from the opening session, David Warner realised the futility of trying to hang around and scored at a run-a-ball rate.
He made 25 before falling lbw to Ramesh Mendis.
Marnus Labuschagne fell for 13 after a rash reverse-sweep shot found the only fielder on the vast off-side and Steve Smith was run out for six after a mix-up with Khawaja.
Khawaja used a review to overturn an lbw decision against him and also survived a missed stumping opportunity as he built up his score.
Earlier, a belligerent 58 by Niroshan Dickwella rescued Sri Lanka who had been on 97-5 in the second session.
Sri Lanka captain Dimuth Karunaratne (28) combined with Pathum Nissanka (23) to give the hosts a decent start after electing to bat.
Off-spinner Lyon was introduced within the first 30 minutes and his first delivery turned sharply, offering a glimpse of what the batsmen can expect in the next four days.
Australia captain Pat Cummins dismissed Nissanka and Alex Carey took the second of his four catches behind the stumps when Kusal Mendis nicked Mitchell Starc.
Warner took a diving catch at gully to send back Karunaratne before Swepson twisted the knife combining drift and turn.
The leg-spinner fired identical loopy leg-breaks to remove Dhananjaya de Silva and Dinesh Chandimal off successive deliveries but was denied a hat-trick by Dickwella.
Angelo Mathews made 39 before walking into Lyon’s leg-slip trap but Dickwella decided offence was the best defence.
The stumper-batsman raced to a 42-ball fifty with the sweep shot, both traditional and reverse, proving particularly productive for the left-hander.
Even more audacious was his ‘Dilscoop’ — a shot named after compatriot Tillakaratne Dilshan — when he bent forward to scoop a Cummins delivery over his own and the wicketkeeper’s heads for a boundary.
Lyon wrapped things up quickly after the tea break, dismissing Ramesh Mendis and Dickwella in successive overs.
Eleven years after his maiden Test five-for on debut in Galle, the off-spinner bagged his 20th at the same venue when Khawaja plucked a stunning catch at wide mid-on.
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