Sri Lanka were 4-98 at tea on day three of the second Test, still 59 runs behind an Australian side that had lost 7-84 before lunch and were all out for 414 in their first dig.
Like the hosts' before them, Australia's spinners weaponised the new ball on a turning Galle wicket, with Lyon (2-27) and Matt Kuhnemann (2-22) finding breakthroughs.
Kusal Mendis is the only out-and-out batter remaining in the Sri Lankan pavilion, leaving Australia strong favourites for a 2-0 series whitewash even if they need to bat again.
Australia have not won every Test match of a series on the subcontinent since a 2-0 sweep of Bangladesh in 2006.
Lyon came up one wicket short of his milestone in Sri Lanka's first innings at the crease, but had his 550th Test scalp shortly after lunch.
The veteran bowled a full delivery around the wicket that Sri Lanka's best batter this series Dinesh Chandimal (12) miscued to a diving Beau Webster at mid-off.
With the catch, Sri Lanka fell to 3-39, 118 runs from equalling Australia's first-innings score and even further from posting a competitive total of their own.
Lyon joined Shane Warne (708) and Glenn McGrath (563) as one of only three Australians to reach 550, and he's all but guaranteed to overtake the second of those men by the end of the calendar year.
Lyon wasn't done there and ended a disappointing series for Sri Lanka's form batter of 2024 Kamindu Mendis (14) when the No.5 chipped to Usman Khawaja at mid-off.
Kamindu only managed to add 10 runs after narrowly surviving an lbw review from Australia that showed more than a little ball on leg stump.
Kuhnemann continued a great series and had an immediate breakthrough skidding into the off-stump of Pathum Nissanka (8) in the second over of the innings.
The 100-match Test career of retiring opener Dimuth Karunaratne (14) ended when he edged Kuhnemann to Alex Carey, with the Australians lining up to applaud the 36-year-old as he left.
Travis Head (0-33) was thrown the ball ahead of debutant Cooper Connolly, whose part-time left-arm finger spin has been used for only three overs so far this match.
Earlier, Prabath Jayasuriya (5-151) reasserted himself as one of Galle's great bogeymen as he pushed the Australians off their perch following a dominant second day.
He had the big wickets of Australia's Alex Carey (156) and Steve Smith (131) within an hour after their 259-run stand that powered Australia into the ascendancy.
Jayasuriya bowled Carey attempting his favourite reverse sweep after dismissing Smith caught behind on his way to a ninth five-wicket haul in Galle.
First-Test centurion Josh Inglis (0), highly touted Connolly (four) and allrounder Beau Webster (31) all missed out on big scores as Australia slipped from 3-330 to  all out for 414.
Carey had earlier brought up 150 runs off only 175 balls, overtaking two knocks of 144 by Adam Gilchrist to set a new high score for Australian wicketkeepers in Asia.
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