Paceman Lee and part-time spinner David Hussey combined to reduce Sri Lanka to 144-6 in the 31st over before Kulasekara (73) and Tharanga (60) gave Australia a scare, firing the tourists to 248 with eight overs to spare.
Kulasekara smashed three sixes and seven fours to leave Sri Lankan fans jumping for joy in the stands, but was out swiping straight to extra cover for Hussey's fourth wicket.
Hussey had a hand in the eighth wicket, snaffling a slogged shot in the deep to dismiss Tharanga for 60 off the bowling of Watson.
Dhammika Prasad battled on with an unbeaten 31 but ran out of partners, with Lasith Malinga smashing straight to Mike Hussey at deep square leg on the second ball of the last over to concede defeat.
In Image: Sri Lanka's Nuwan Kulasekara plays a shot