Set an improbable 507 runs to win the Test and square the two-match series, the Sri Lankans reached tea at 139 for one.
Atapattu was unconquered on 74 in 157 minutes and Sangakkara was not out 51. The pair batted through the entire middle session in an unbroken partnership of 124 runs.
The Australian bowlers were finding it difficult to dislodge the experienced pair on a benign Bellerive pitch under sunny skies with four sessions left to get a result after romping to an easy win in the first Brisbane Test.
The tourists began disastrously when opener Michael Vandort (4) badly miscued an attempted pull shot off Mitchell Johnson and skied to Rhett Lockyer, a fielding substitute for injured Andrew Symonds.
Australian skipper Ricky Ponting, who remained unbeaten on 53 with Mike Hussey on 34, made the declaration 45 minutes before lunch giving his bowlers just over five sessions to bowl out the Sri Lankans.
The Sri Lankans will have to create history to win the Hobart Test.
The world record fourth innings run chase for victory was 418 for seven by the West Indies against Australia in Antigua in May 2003.
The highest winning run chase at Bellerive was Australia's 369 for six against Pakistan in November 1999 -- which ranks as the fourth highest in Test history.
Opener Phil Jaques was the only Australian wicket to fall on Monday, caught on the point boundary by Michael Vandort off Lasith Malinga for 68.
Jaques was chasing his third consecutive century against Sri Lanka in the series.