Sri Lanka (257, 8-211) went to stumps on day three of the second Test only 54 runs ahead of Australia (414) in their second innings, with Kusal Mendis (48 not out) their last recognised batter.
Australia could seal the win in the morning session on Sunday, but won't want the run chase getting out of hand given the tricks the Galle pitch played early on Saturday.
"There's a big job to do in the morning with the ball," said spinner Nathan Lyon, after a day when he became only the third Australian to pass 550 Test wickets.
"It's been a hard toil today, there's no point in hiding behind that. But I'm very proud of our bowling group."
It was a whirlwind third day in sunny Galle, where Australia lost 7-84 in a little less than two hours as Prabath Jayasuriya (5-151) weaponised the new ball.
In reply, Lyon (3-80) led the way as Australia pushed Sri Lanka to 4-81, still 76 runs from drawing level.
At that stage, Australia might have thought they'd only need until the end of the day to complete a 2-0 series whitewash - a feat they haven't achieved in Asia since touring Bangladesh in 2006.
But after tea, veteran Angelo Mathews (76) and Kusal partnered in a 70-run stand that erased an Australian lead built up by Steve Smith (131) and Alex Carey (156).
After surviving some early punishment from Lyon, Mathews drove Matt Kuhnemann (4-52) over long-off for six to bring up his half-century off 76 balls.
Australia had no answers for the combination, with Steve Smith unsuccessfully turning to Mitch Starc (0-22) and part-timer Cooper Connolly (0-9) under the fading light.
Beau Webster proved the unlikely hero, first diving low to dismiss Mathews when the veteran holed out to an ambitious slog sweep from Lyon's bowling.
Sri Lanka were into the bowlers when Webster (1-6) swapped the medium pace of the first innings for the off-spin of earlier in his career.
The towering Tasmanian enticed Ramesh Mendis into edging to short leg, and Sri Lanka's hopes of a big total appeared a distant memory with the bowlers at the crease.
Kuhnemann, Australia's best bowler of the match, put Australia on course for victory when Jayasuriya (six) edged him to the slips on the last ball of the day.
Jayasuriya estimated Sri Lanka needed around 100 more runs to keep themselves in the fight.
"They have to play in the fourth innings, it's not easy to win in Galle. I think definitely we are in the game," he said.
Earlier, Lyon joined Shane Warne (708) and Glenn McGrath (563) as one of only three Australians to reach 550 Test scalps when Dinesh Chandimal (12) hit him to a diving Webster.
He is already onto 552 at stumps, and is all but guaranteed to overtake McGrath by the end of the calendar year.
Kuhnemann continued a great series and had an immediate breakthrough skidding into the off-stump of Pathum Nissanka (eight) 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.
Kuhnemann's sparkling series has made it easy to forget the 28-year-old dislocated his right thumb only two weeks before the first Test.
"I know it's hurting him, I know that. But he's there showing a lot of courage," Lyon said.
In Australia's innings, Jayasuriya had the big wickets of Carey and Smith within an hour of the restart 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 Webster (31) all missed out as Australia slipped from 3-330 to all out for 414.
Carey had earlier brought up his 150 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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