Button's compatriot and team mate Lewis Hamilton, who started on pole, finished third to round out the podium while Vettel's team mate Mark Webber was fourth, his best ever finish in his 11th attempt to win his home race.
"Oh fantastic, what a great way to start the year, guys! Seriously, fantastic!" former world champion Button said on the team radio. "The car is beautiful and she's quick. Thanks guys."
Vettel, who won 11 of 19 races last year including an absolutely dominant victory at Albert Park, benefited from a mid-race safety car to get past Hamilton but conceded he had been very much second best.
"I think Jenson was unbeatable today," he told a news conference. "I'm very happy with second, it's a lot of points. They are the ones to beat at the moment and we'll see what happens next week."
Fernando Alonso drove a spirited race to finish fifth for troubled Ferrari ahead of the Sauber of Japan's Kamui Kobayashi after Williams driver Pastor Maldonado crashed out spectacularly on the final lap while running sixth.
"After yesterday's qualifying to be fifth behind the two Mclarens and two Red Bulls is a pretty good result in terms of number of points," said Vettel.