Dimuth Karunaratne's Test career ended in a whimper as Sri Lanka's most prolific Test opener fell caught behind to Kuhnemann after making 14.
Angelo Mathews's gutsy 76 helped Sri Lanka to avoid an innings defeat in the second Test in Galle on Saturday but Australia remained on course for a 2-0 series sweep.
Having conceded a first-innings lead of 157 on a track dominated by spinners from both sides, Sri Lanka risked a second successive innings defeat in the series when they slumped to 81-4.
Mathews produced a battling half-century to revive Sri Lanka, who finished day three on 211-8 -- effectively 54 ahead with two second-innings wickets in hand.
Kusal Mendis was batting on 48 after the Australian spin duo of Matthew Kuhnemann (4-52) and Nathan Lyon (3-80) ran through the Sri Lankan batting order.
Earlier, resuming on 330-3, Australia lost their remaining seven wickets in the morning session to be all out for 414 in their reply to Sri Lanka's first innings of 257.
After four quiet overs, Prabath Jayasuriya rattled the tourists by claiming two wickets in three deliveries.
Steve Smith's stellar knock of 131, which included 10 fours and a six, ended when he fell caught behind to the left-arm spinner.
Two balls later, Josh Inglis fell for a duck, his stump pegged by a Jayasuriya delivery that skidded and took an inside edge.
Jayasuriya also ended Alex Carey's career-best 156, which included 15 fours and two sixes, by bowling him after the batter had fluffed his pull shot.
Carey became the first Australian wicketkeeper to score 150 in Asia.
Jayasuriya also bowled Mitchell Starc to register his ninth five-wicket haul at his happy hunting ground.
Having conceded a significant lead on a turning track, Sri Lanka wobbled early in their second innings.
Pathum Nissanka (eight) hit Starc for two fours in the first over but lost his off-stump to Kuhnemann's third delivery in the next over.
Dimuth Karunaratne's Test career ended in a whimper as Sri Lanka's most prolific Test opener fell caught behind to Kuhnemann after making 14.
Lyon removed Dinesh Chandimal and Kamindu Mendis and Smith took a stunning low catch in slip to remove his Sri Lanka counterpart Dhananjaya de Silva (23).
Mathews combined with Kusal to help Sri Lanka erase their deficit.
Lyon broke the dogged 70-run stand when Mathews swept him to Beau Webster, who took a tumbling catch at backward square leg.
Bowling off-spin, Webster removed Ramesh Mendis for a duck and Kuhnemann removed Jayasuriya with the final delivery of the day.
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