There was drama all over the track except at the very front, where Button remained from the first corner where he passed team-mate Lewis Hamilton, until the chequered flag.
Sebastian Vettel demonstrated the race pace of the Red Bull to take second place, while pole-sitter Hamilton was left to rue some bad luck with pit stops to end up in third.
Mark Webber was fourth in his home Grand Prix - his best result in Australia - while Alonso took a surprise fifth in the struggling Ferrari.
Pastor Maldonado, breathing down Alonso's neck in the final laps, looked set to finish a creditable sixth place for Williams, but crashed just a few corners from the end to spark a frenzied race to the line for the lower points positions.
Kamui Kobayashi pounced for sixth, while Kimi Raikkonen stole a couple of places to take seventh, with Sergio Perez, Australian Daniel Ricciardo and Scot Paul Di Resta rounding out the top 10.
Having taken the lead, Button never looked back, opening up an advantage that gave him breathing room over his team-mate.
Webber reprised his starting woes from 2011, being clipped twice in the race to the first corner and slipping down to ninth, while Romain Grosjean also dropped down sharply from third on the grid and soon exited the race after a clash with Maldonado.
Another man forced out early was Michael Schumacher, who had been running in third before technical problems put paid to his chances after 10 laps.
Alonso belied the concerns over the new Ferrari by jumping from 12th to fourth and matching the pace of the Red Bull ahead of him in the early stages of the race, while the Saubers and Raikkonen also made progress from humble starting positions on the grid.
Though Hamilton did not get the best of starts, he was unfortunate with both his pit stops. On the first he emerged behind Perez, allowing Button to stretch the lead to 10 seconds. Having avoided that error on the second, Vitaly Petrov's retirement on the home straight forced the safety car out on the next lap, meaning that Vettel was able to dive in to the pits immediately and leapfrog him into second place.
When the safety car returned into the pits with 17 laps of racing left, it looked like Button would be faced with a challenge from the world champion.
But he opened a three-second advantage over Vettel in scarcely a lap, and never looked back, taking the 13th race win of his career and his third at Albert Park.