Sri Lanka were set 152 for victory from 41 overs under the Duckworth-Lewis method after Australia had been bowled out for 158 in an innings that was interrupted for two hours by wet weather.
They never looked like needing much more than half of their allocated overs, however, and captain Mahela Jayawardene smashed an unbeaten 61 as they reached their target for the loss of two wickets with 101 balls to spare.
Sangakkara delighted the large Sri Lankan contingent in the crowd by becoming the 10th cricketer to pass the 10,000-run milestone on his way to a 29-ball 30. He was congratulated at the crease by Jayawardene, another member of the exclusive club.
Opener Tillakaratne Dilshan made the other major contribution to the Sri Lanka tally with a quickfire 45 before being caught by David Hussey off Clint McKay, handing out brutal punishment to Brett Lee in particular early in the innings.
All rounder Thisara Perera was the standout for Sri Lanka in the field, taking 2-29 with his right-arm medium pace and running out two more Australian batsmen with brilliant fielding.
The 22-year-old's contribution would have been more telling had Lasith Malinga managed to hold onto a catch from David Hussey off his first ball of the day, when the batsmen was on eight.
Hussey was also dropped by Angelo Mathews when he was on 33 and he made the most of his reprieves to score 58, his third half century of the series.
His stand of 49 for the ninth wicket with Mitchell Starc rescued the Australian innings from 104-8 after a top order collapse but ultimately it proved too little to save the hosts from defeat.
Ricky Ponting - standing in as captain for the injured Michael Clarke - had a miserable evening after winning the toss, continuing his poor form with the bat by being caught and bowled by Farveez Maharoof for two.
It was Sri Lanka's first win in the series, which continues when the hosts take on India in Brisbane on Sunday.
