The German started second on the grid, and though Lewis Hamilton nailed his start, the Ferrari was able to trail in close quarters for the first stint.
By lap 17, however, Hamilton was struggling for grip on his used ultrasoft tyres, which prompted the Mercedes pit wall to call him in for an early change to the soft compound.
The strategy backfired, however, because Hamilton emerged behind Max Verstappen’s Red Bull Racing car encumbered by its own set of used ultrasofts.
While Hamilton languished behind his slower rival, Vettel was able to extend his lead, and when Ferrari stopped the German for his sole tyre change, he emerged ahead of Verstappen and Hamilton – and in effective lead of the grand prix.
Vettel wasted no time rebuilding his gap, and by the time Verstappen stopped for new tyres, the gap between the Ferrari and Hamilton was already up to eight seconds.
“Forza Ferrari!” Vettel exclaimed after the race in Italian. “This is for all of you. Thank you. Great work, this is a great car.”
It was an immensely popular win, with Australian fans celebrating hard Ferrari’s first win in Melbourne since 2007 despite a dearth of overtaking.
“Thank you everyone,” Vettel said on the podium. “It’s what we needed. The team’s been working very, very hard. The car’s been behaving well, and it’s incredible fun to drive.”
Few believed Ferrari would be in a position to win races this season after a lacklustre 2016 and turmoil behind the scenes, and the four-time champion acknowledged the hard work it took to ensure the team could find itself at the pointy end of the field.
“For now I think we’re just over the moon. We’re just happy.
“As I said, it’s been a hard winter and an incredible win today. I was really trying to keep the pressure on to make sure they hear the message that we’re here to fight.”
Hamilton was unable to activate the soft tyres as effectively as Vettel, and the Briton also attributed some of his deficit to an unknown power unit inconsistency that prevented him from mounting a challenge for the lead.
“Firstly a big congratulations to Ferrari and Sebastian,” Hamilton said. “A great weekend for the team, but in the race we just struggled with the tyres. I just ran out of grip.
“Sometimes that’s the way it goes, [but] overall it’s a great race [and] great points for the team.”
Valtteri Bottas finished third and only two seconds behind his new teammate for Mercedes.
Bottas hovered behind Hamilton for most of the second half of the race, threatening to usurp the senior Mercedes driver on a number of occasions, but Hamilton managed to respond to the Finn’s quickest laps sufficient to hold his advantage.
“Obviously this is my first race with the team, and I really want to thank everyone,” Bottas said. “We need to work hard, and we’re definitely ready for that – starting with a podium, we can improve it for here.”
Toto Wolff suggested Mercedes had misread the race on an afternoon the car didn’t have the advantage the team has ordinarily enjoyed.
“It’s always frustrating when you lose,” Wolff told the BBC. “You just need to be faultless in order to win.
“Today we weren’t quick enough, and the tyre was degrading and maybe we thought it was degrading more than it was.
“It was a good wake-up call.”
Kimi Räikkönen drove an anonymous race in fourth, with most of his efforts spent defending against Max Verstappen during the second half of the race.
The Dutchman couldn’t find a way through, though, and had no choice but to settle for his grid position of fifth.
Felipe Massa drove a lonely race for Williams in sixth, finishing more than 50 seconds behind Verstappen and almost 30 seconds ahead of everyone else.
Sergio Perez in his pink Force India defended his seventh place against a late-charging Carlos Sainz, who led Toro Rosso teammate Daniil Kvyat home in eighth and ninth.
Esteban Ocon made an aggressive overtaking move in the second Force India on Fernando Alonso for the final point in the dying laps of the race, which would have denied McLaren its first point were it not for the Spaniard’s retirement one lap later.
Renault’s Nico Hülkenberg, Sauber debutant Antonio Giovinazzi and McLaren’s Stoffel Vandoorne failed to score points.
The Australian Grand Prix was a race of attrition, with seven cars failing to take the chequered flag.
The highest profile retirement was home crowd favourite Daniel Ricciardo, who suffered a gearbox problem on his way to the grid.
Red Bull Racing recovered his car and sent it onto the track on the second lap of the race, but 28 laps later the Australian’s car failed again, this time with a suspected power unit failure.
“It’s frustrating obviously,” Ricciardo lamented to the BBC. “Not the weekend I wanted at home – that’s probably the more frustrating thing.
“The boys had a long night trying to fix the car. It’s just bad luck I guess.”
The 2017 Formula One season continues with the Chinese Grand Prix on 9 April.