Qualifying resumed at 1100 local time on a wet track and Rosberg continued his impressive form from Saturday to set the pace in second qualifying.
McLaren, who have been struggling for pace this weekend, made a mistake in switching both their cars to slicks too early.
But while their new driver Sergio Perez stayed on them and was knocked out in the second session, Button decided to come back in for intermediate tyres.
The decision ensured he progressed to the top 10 shoot-out but the poor performance of the car ensured he could manage only 10th on the grid, nearly three seconds off Vettel. Perez will line up 15th.
The dry conditions at the end of qualifying allowed Red Bull to prove the dominant form they had showed in Friday practice was no illusion.
Vettel was a second clear of the field on the first runs in final qualifying, which were done on intermediate tyres.
On slicks at the end of the session, Vettel was 0.68secs faster than Hamilton and more than a second ahead of Massa.
He could potentially have gone faster on his final lap before deciding to abort it after the team told him he could not be beaten.
Australian GP: Sebastian Vettel's pole lap
"Definitely not easy in these conditions," Vettel said.
"I had two good runs, one on inters and one on dries. The car felt very good, similar to Friday.
"It is pretty difficult to find the limit in these conditions but I was happy with the lap. As soon as I got confirmation that it was enough I pulled in to save the tyres."
The final six positions on the grid were decided on Saturday, before qualifying was postponed.
In the first session, Hamilton had had a narrow escape when he spun at Turn Two and his Mercedes initially appeared to be beached on the edge of track - the rear wheels were spinning uselessly as he tried to rejoin.
But Hamilton was able to reverse back onto the circuit and he ended up 10th fastest despite a damaged rear-wing endplate.
Hamilton, who was significantly slower than Rosberg when the track was wet on both Saturday and Sunday, said: "It was very tricky out there but I'm very happy I could get the job done. To be this far up coming into the season... I said yesterday it was a blessing they moved it to today and it really was."
Williams driver Pastor Maldonado and Sauber's rookie Esteban Gutierrez were the two drivers to be knocked out in the first part of qualifying along with the Marussias and Caterhams.
Both Marussias outqualified both Caterhams, and Frenchman Jules Bianchi continued the good impression he has made since being taken on by Marussia at the final pre-season test.
Bianchi was 0.767secs quicker than team-mate Max Chilton.
The race is a huge unknown - no-one knows how the tyres will last or how strategy will unfold.
Qualifying top 10
1. Sebastian Vettel - Red Bull - 1:27.407
2. Mark Webber - Red Bull - 1:27.827
3. Lewis Hamilton - Mercedes - 1:28.087
4. Felipe Massa - Ferrari - 1:28.490
5. Fernando Alonso - Ferrari - 1:28.493
6. Nico Rosberg - Mercedes - 1:28.523
7. Kimi Raikkonen - Lotus - 1:28.738
8. Romain Grosjean - Lotus - 1:29.013
9. Paul di Resta - Force India - 1:29.305
10. Jenson Button - McLaren - 1:30.357
Australian Grand Prix, day three
Sunday, 17 March: Qualifying highlights: 05:15 GMT, BBC One. Race: 06:00 GMT, 05:30 GMT on 5 live and live text commentary online. Race highlights: 14:20 GMT, BBC One.
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