bowler Ben Hilfenhaus batted despite a side strain.
Australia collapsed, losing their last eight wickets for 146, after openers David Warner (68) and Ed Cowan (56) put on 132 for the first wicket.
Left-arm spinner Rangana Herath captured five for 95 to lead the Sri Lanka counter-attack.
The highest successful run chase in 10 previous Tests at Bellerive was Australia's 369 for six against Pakistan in 1999.
The home side looked to be travelling along nicely going to lunch at 146 for two before it all unravelled in the middle session as they went after quick runs for an apparent innings declaration.
Shane Watson went three overs after lunch, going after a referral went against him for a stumping off Herath for five and he was soon followed by Phil Hughes.
Hughes, who scored 86 in the first innings, chopped Shaminda Eranga on to his stumps for 16.
Matthew Wade, sent in ahead of Clarke to accelerate the scoring, hit 11 off 14 balls before he holed out to Nuwan Kulasekara at long on off Herath.
Clarke went after the runs and reached a whirlwind half-century off 43 balls, but he called for the team physio for a right hamstring injury and retired hurt on 57.
The wickets kept tumbling and Peter Siddle was brilliantly snapped up by a diving Prasanna Jayawardene off Welegedara for four.
Mitchell Starc hit five off three balls before he was leg before wicket to Welegedara as the last able-bodied Australian player Nathan Lyon came out to bat with the lead over 370.
Lyon added 11 off as many balls before he became Herath's fourth victim of the innings and Ben Hilfenhaus came out to bat despite being inconvenienced by a side strain.
Hilfenhaus lasted eight balls before he became Herath's fifth wicket when lbw for a duck.
The afternoon clatter of wickets was in sharp contrast to the extended morning session when Warner and Cowan were largely untroubled.
Warner turned on the fireworks to slog Herath for six over long-on and then switch hit the left-arm spinner for four.
But he perished when he attempted to cut Herath and feathered a catch to wicketkeeper Prasanna Jayawardene on 68 ending a 132-run opening stand.
Cowan was bowled off-stump by Welegedara for 56 shortly before lunch with one that cut back in.