Sri Lanka's Dinesh Chandimal was waging a lone battle on 63 with Australia ahead by 518 runs and tightening their grip on the first test before rain forced early lunch in Galle on Friday.
The hosts were 136-5 by the first break on day three with wicketkeeper Kusal Mendis also at the crease on 10 after losing two more wickets overnight.
Kamindu Mendis departed for 15 after an edge to the keeper off Mitchell Starc, further denting the Bradman-like average he maintained at the start of his test comeback last year.
Captain Dhananjaya de Silva looked comfortable in his brief stay before rushing down the track to Matthew Kuhnemann and missing the ball completely, gifting Alex Carey a regulation stumping and leaving for 22.
Left-arm speedster Starc expertly created rough patches outside the off-stump while Nathan Lyon relentlessly probed away from the other end.
Chandimal, who was promoted to the crucial No 3 position last year and has averaged over 50 since, stood firm and bunkered in for an uphill battle in avoiding the follow-on.
Australia declared on 654-6 late on the second day after opener Usman Khawaja's career-best 232.
The opening day saw stand-in skipper Steve Smith join the elite club of 10 000 career test run-makers, the fifteenth player and fourth Aussie to the milestone.
Picturesque Galle will also host the second and final match of the series next week before a two-match ODI series in Colombo.
AUSTRALIA: Usman Khawaja, Travis Head, Marnus Labuschagne, Steve Smith (capt), Josh Inglis, Beau Webster, Alex Carey (wk), Mitchell Starc, Matt Kuhnemann, Nathan Lyon, Todd Murphy.
SRI LANKA: Dimuth Karunaratne, Oshada Fernando, Dinesh Chandimal, Angelo Mathews, Kamindu Mendis, Dhananjaya de Silva (capt), Kusal Mendis (wk), Prabath Jayasuriya, Nishan Peiris, Asitha Fernando, Jeffrey Vandersay.