The wicketkeeper-batsman reached three figures with a well-run two off the final ball of England's innings, after the visitors were put into bat.
When Buttler came to the crease in the 23rd over, England were precariously placed on 107-4 at the SCG.
The 27-year-old started patiently and his first half-century took 52 balls, while the next 50 took just 31 balls.
At the other end, Chris Woakes carried on from his match-winning showing at the Gabba to post 53 from just 36 balls, the duo hitting 102 off the last 10 overs.
However, Woakes conceded just five runs and took the vital wicket of Stoinis to seal a superb victory for England.
Eoin Morgan's side now have an unassailable 3-0 lead in the five-match series, with the fourth ODI at Adelaide on Friday.