Kumar Sangakkara and Mahela Jayawardene kept the Australians at bay till the end of play after losing openers cheaply.
Hobart: Australia lead Sri Lanka by 328 runs heading into the final day of the first Test, setting the tourists a challenging victory target before restricting them to a second-innings 65-2.
Australia captain Michael Clarke helped his side to 278-9 and a target of 393 before Shane Watson and Mitchell Starc removed the Sri Lanka openers before close of play Monday.
Kumar Sangakkara (18) and Mahela Jayawardene (5) were the unbeaten batsmen, intent on staying at the crease rather than scoring. Jayawardene faced 25 balls before his first run.
The highest successful run chase in 10 previous Tests at Hobart was Australia's 369-6 against Pakistan in 1999. Sri Lanka has never won in Australia and has never successfully chased more than 352.
Clarke retired with a right hamstring injury and Cricket Australia said he will be evaluated by medical staff, but the skipper led his team onto the field for the start of Sri Lanka's second innings after tea, taking a fielding position at first slip.
The fifth-day pitch at Bellerive Oval should firmly favor the home bowlers, who have already extracted plenty of movement and seen the ball stay low. Sangakkara in particular was fortunate to see a low-bouncing delivery from Watson that entirely beat him miss his stumps.
Watson snared Dilshan, who scored a crucial first-innings century to keep Sri Lanka in the match, for just 11 with his first ball — finding an outside edge that went to Matthew Wade behind the stumps.
It was 47-2 when Starc beat Karunaratne with a perfect Yorker, just as he bowled Dilshan in the first innings.
The pace then slowed to a crawl as the experienced Sangakkara and captain Jayawardene added just 18 runs in the remaining 13.5 overs before stumps.
The imposing target was set up by a pair of crucial partnerships in the Australia batting lineup. Openers David Warner and Ed Cowan resumed the day at 27-0 and took the score to 132-0 before Warner became the first of Sri Lanka spinner Rangana Herath's five victims, caught behind for 68.
Cowan departed for 56 soon after with the run rate dropping as three middle-order wickets fell for 35 runs after lunch.
Clarke and Mike Hussey picked up the pace with an aggressive 57 runs from 56 balls, Clarke making 57 from 46 balls before retiring hurt.
Hussey enjoyed a 38-ball 31 before running out of partners at the other end.
Herath finished with figures of 5-96 to move ahead of England's Graeme Swann as test cricket's leading wicket taker for 2012. Herath has needed just nine tests for his 60 wickets, three fewer than Swann has played.
Sri Lanka must now hit its highest ever winning total if it is to record a first win from 11 attempts on Australian soil. The West Indies made the highest successful run chase against Australia with 417-8 at St. John's in 2003.