The Englishman led throughout and team-mate Rubens Barrichello completed a Brawn one-two after a dramatic finish.
The Brazilian was promoted after Red Bull’s Sebastian Vettel and BMW’s Robert Kubica collided while disputing second place with three laps to go.
Toyota’s Jarno Trulli was hit with a 25-second penalty after the race giving Lewis Hamilton an unexpected third.
Claiming an eventual podium finish from 18th on the grid was a superb achievement for the British world champion, given the woeful performance of his McLaren coming into the race.
But all eyes were on Brawn after the chequered flag, with the success of the former Honda team – whose survival was only secured by team boss Ross Brawn in early March – firmly capturing the imagination of the sport as it enters a fresh era under new rules.
Victorious Button secured the second win of his 154-race Formula 1 career after driving coolly throughout, showcasing Brawn’s speed to keep Red Bull’s Sebastian Vettel at bay for the vast majority of the race to eventually cruise to a finish behind the safety car.
Button leads the field at the start of the race in Melbourne
"This is a fairytale ending for the first race," said Button.
"Some people may say its a pity the race finished under the safety car but I don’t care, I won the race and that’s all I care about."
Vettel looked set for a brilliant second place but he and Kubica tangled on Turn Three with just three laps remaining.
The Red Bull driver was slow through the first two corners, allowing Kubica to get alongside on the outside.
The Pole gave Vettel room, but the Red Bull’s front wheel tagged the rear wheel of the BMW.
That tipped both into a spin, and damaged their front wings.
Both crashed further around the lap, and although Vettel tried to continue on three wheels he was eventually forced to retire.
Along with earning the Red Bull driver a post-race 10-place grid penalty for next weekend’s Malaysian Grand Prix, the crash brought out the safety car for the second time in the race and it stayed at the head of the field until the end.
Brawn’s one-two looked unlikely at the start of the race as Barrichello, sat next to Button on the front row, was easily passed by the chasing pack as his anti-stall system kicked in.
The Brazilian also tangled with Ferrari’s Kimi Raikkonen to cause significant damage to his front wing, which had to be replaced during his stop on lap 18.
The frenetic start also saw Red Bull’s Mark Webber collide with McLaren’s Heikki Kovalainen, causing the Finn to retire and putting the Australian to the back for the duration of his home race.
Button built a lead of nearly 4.5 seconds in the first two laps but Vettel stabilised the gap, which stayed at between four and five seconds until the first pit stops.
Kubica – one of several drivers to start the race on the slower, softer tyres in order to get them out of the way early – stayed in the hunt while Hamilton continued to make up good ground after a brilliant first lap which saw him up to 12th from 18th on the grid.
The first of two safety car periods happened on lap 19, following a crash by Williams driver Kazuki Nakajima.
When it pulled in again, Button again built a five-second lead over a few laps before Vettel held it.
But both – now on the softer, slower tyres – came under pressure late in the race from Kubica, who was now benefiting from the team’s decision to use the faster tyres late on.
That pressure on Button was punctured in an instant as the Pole collided with Vettel allowing Trulli and Hamilton, whose McLaren had woefully underperformed in qualifying by his own admission but was now seemingly capable of giving him a drive, to reap the rewards.
Hamilton received a further boost following the race when he was elevated to third after veteran Italian Trulli was handed a 25-second penalty by the stewards for passing Hamilton under yellow flags following the crash involving Vettel and Kubica.
Toyota’s Timo Glock finished fifth in Melbourne ahead of Renault’s Fernando Alonso and Williams’s Nico Rosberg, who suffered a sticking front wheel at a pit stop on lap 15 which scuppered his chance of a podium finish.
Red Bull’s Sebastien Buemi made an impressive debut to secure one point in his first ever F1 Grand Prix.
And Ferrari finished without a point in the season opener for the second year running, as Felipe Massa – running in third place – sustained a reliability failure on lap 45, three laps after team-mate Raikkonen spun out.