Vettel held off reigning champion Lewis Hamilton to take the checkered flag in Melbourne, with his Ferrari teammate Kimi Raikkonen in third place.
Vettel, who was repeating his 2017 success at Albert Park, was able to leapfrog polesitter Hamilton while a virtual safety car was deployed on lap 26 of 58 at Albert Park.
That came about as both the Haas cars of Kevin Magnussen and Romain Grosjean retired in quick succession after pit stop errors left them driving with loose wheel nuts.
Their despair was clear to see as both were set for solid points-scoring finishes, but it left Vettel with an opportunity.
He came in for his pit stop and emerged in front of Hamilton, who up to that point had controlled the race from the front.
"What just happened, guys? Why did you not tell me Vettel was in the pits?," a shocked Hamilton asked his Mercedes team over race radio.
"Was that my mistake? Was I supposed to go quicker through the safety car?"
"We thought we were safe, but there is obviously something wrong," came the response.
Mercedes later admitted that a software error had led to a miscalculation in the speed Hamilton was allowed to go under the virtual safety car.
In an unexpected lead, Vettel was able take advantage of fresher tires and the difficulty of overtaking on the street circuit to secure his 48th F1 career victory.
Home favorite Daniel Ricciardo pushed Raikkonen all the way to take fourth place for Red Bull, with two-time former world champion Fernando Alonso an excellent fifth for an improved McLaren team.
Ricciardo's teammate Max Verstappen never fully recovered from an early race spin and had to settle for sixth spot.
"We got a bit lucky obviously with the safety car," said Vettel.
"My start, I was hoping to be a bit better, didn't really work, so I had to settle for third. I was praying for a safety car."
A disappointed Hamilton admitted: "Today they did a better job. We've got to go back to the drawing board and work on it.
"But we still have great pace. This is an awesome circuit, but it's so hard to overtake."
Hamilton had led from pole after a clean start with Raikkonen staying ahead of Vettel in the early running.
The virtual safety car overtakes turned the race order on its head, with Alonso also taking advantage.
"Well done guys, very proud of you. Long winter, long seasons in the past but now we can fight, we can fight," the Spaniard said over his team radio.
Vettel also showed his experience by easily holding off Hamilton, finishing five seconds clear as the Briton conserved his engine ahead of the second round in Bahrain in a fortnight.
The result mirrored last year with Vettel making most of the early running in the title race until Hamilton dominated the latter part of the season to win his fourth world championship crown.
The arch-rivals are both battling for a fifth global title in what looks set to be a compelling 2018 season with Red Bull also showing good pace in the early tussles as Ricciardo set the fastest lap in his attempt to make the podium in front of his home fans.
The Haas team, ironically now supplied with Ferrari engines, was also putting up a stellar showing with Magnussen and French driver Grosjean running fourth and sixth until their respective mishaps.
But both their pit stops were flunked as first Denmark's Magnussen pulled off at Turn Three on lap 22, with Grosjean stopping two laps later.
Grosjean left his car on the straight between Turns Two and Three, leaving officials to initially impose a virtual safety car, where the drivers are asked to maintain a set speed and maintain the gaps between them.
After Vettel, who had delayed his initial pit stop, had taken the lead it became a full safety car period, but with the German able to start from the front he was never headed.