Sri Lanka 257 & 211 for 8 (Mathews 76, K Mendis 48*, Kuhnemann 4-52, Lyon 3-80) lead Australia 414 (Carey 156, Smith 131, Jayasuriya 5-151) by 54 runs
Sri Lanka's hopes of setting a tricky chase faded late on day three as Beau Webster provided key contributions with the ball and in the field to inch Australia close to a series victory.
Angelo Mathews and Kusal Mendis had breathed life into the second Test with an outstanding 70-run partnership as Sri Lanka's lead grew to 41 runs. But less than 30 minutes before stumps, Mathews lost his concentration on 76 and swept Nathan Lyon to fine leg, where a diving Webster completed a terrific catch.
In another masterstroke from stand-in captain Steven Smith, Webster was handed the ball and unfurled his offspin - instead of seam - to dismiss Ramesh Mendis for a duck.
Webster almost had Prabath Jayasuriya caught and bowled first ball, while Mendis survived a huge lbw shout off Lyon that was given not out by umpire Joel Wilson. Australia unsuccessfully reviewed in an umpire's call, but Smith's gut instincts again proved correct when left-arm spinner Matthew Kuhnemann was tasked with bowling the final over and he promptly dismissed Jayasuriya.
Sri Lanka lead by just 54 runs with two wickets in hand having lost 3 for 13 in the last five overs. Their chances of victory have effectively evaporated unless Mendis, who finished unbeaten on 48 having been left stranded on 85 in Sri Lanka's first innings, can produce a miracle.
Sri Lanka had earlier appeared headed for a defeat within three days. Trailing by 157 runs on the first innings, they stared down the barrel of a quick demise at 39 for 3 when Lyon claimed his 550th Test wicket after dismissing Dinesh Chandimal for 12.
Retiring Dimuth Karunaratne had fallen for 14, caught behind off Kuhnemann, bringing down the curtain on his 100-Test career. After prolonging his last moments at the crease with a review in vain, Karunaratne received an embrace from Lyon and Chandimal before walking off to a big ovation from those at the ground, including the Australians.
It was Kuhnemann's second wicket after he bowled opener Pathum Nissanka with a good length delivery that slid on.
Chandimal had scored a couple of 70s in this series, but needed an even bigger contribution if Sri Lanka were to turn around their dire situation. But after being pinned down, Chandimal lost his patience against Lyon and hit to mid-off, where Webster moved nimbly to take a sharp catch low down.
Mathews and Kamindu Mendis provided some resistance with a 42-run partnership having both received reprieves after missed chances by Alex Carey with the gloves.
Entering the series averaging over 70, Kamindu's tame series ended when he chipped Lyon to mid-off but skipper Dhananjaya de Silva batted crisply in a 47-run partnership with Mathews.
They moved along with relative ease to chip away at the deficit and Mathews reached his half-century in style with a six over cover off debutant Cooper Connolly, who was used for a couple of overs.
Just when Sri Lanka's hopes were raised, Smith seemingly dealt them a crushing blow when he completed a sensational one-handed, diving catch at slip after de Silva edged Kuhnemann.
But Mendis came out with an aggressive mindset to rattle a tiring Australia attack amid the humidity. He levelled the scores with a slog sweep off Lyon that sailed into the crowd before lifting Sri Lanka into the lead on the next ball.
Sri Lanka were having arguably their best period with the bat in the series before Mathews' rush of blood. It was a disappointment for Sri Lanka after their best day of the series ended on a sour note.
Sri Lanka's attempts at a fightback started when they claimed seven wickets in the morning session to bowl out Australia for 414. Jayasuriya found venomous bite with the second new ball to pick up four of his five wickets in the session.
Smith and Carey's magnificent partnership of 259 ended less than 20 minutes into the day's play. In another batting record for Australia in this series, it was the highest fourth-wicket partnership by a visiting pair in Sri Lanka in Tests, bettering the 258-stand between Michael Hussey and Shaun Marsh in Pallekele in 2011.
Australia resumed in a powerful position at 330 for 3, but they knew there was work to do in their bid to only bat once in this match. Carey overtook his highest first-class score of 143 in style with a firm sweep that rocketed to the boundary and he also moved past Adam Gilchrist's 144 as the highest score by an Australian wicketkeeper in Asia.
Jayasuriya was rewarded for his attacking line and lengths when Smith's stout defence was finally breached on 131 by a brilliant delivery that pitched on off stump and gripped off the surface. Smith walked off immediately after feathering an edge and he received a rousing ovation, but his dismissal reinvigorated Sri Lanka and brought a different complexion to the game.
Josh Inglis made his way to the crease one spot lower in the batting-order after spending time off the field during Sri Lanka's first innings because of a back spasm. After scoring a memorable debut ton in the first Test, Inglis' stumps were rattled on his second delivery when he played back to a full delivery that skidded on.
Carey on 156 finally succumbed to the sweep, a stroke he had been almost faultless through the innings, when he played over the top of a Jayasuriya delivery and was clean bowled.
All eyes were on Connolly, who boasted a first-class average of 61.80 from four matches but three of those was at the pace-friendly WACA ground.
Connolly has proven to be a big-game performer in his fledgling career, backing his aggressive instincts, and he got off the mark in trademark cavalier fashion with a boundary albeit off a top-edge from a full-blooded sweep.
But Connolly didn't add to his tally and was too ambitious when he skipped down to Peiris only to slice to backward point in an ugly dismissal. Webster provided a steadying 31 and he was to have a bigger impact later in the day's play.
