Vettel, who at 23 became the youngest world champion last year, was too quick for his rivals around the Albert Park street circuit, beating McLaren's 2008 world champion Lewis Hamilton by more than 22 seconds with Russian Vitaly Petrov third in a Lotus Renault.
It was Vettel's 11th win in his 63rd GP and continues from where he left off when he won the final race of last season in Abu Dhabi.
"All in all it was a very good race, there were a lot of things to learn today and we move forward from here," Vettel said.
"I don't think it was an easy race, the start was crucial and being on the clean side of the track I had a very good getaway.
"The second part of the race I didn't know what was going on behind Lewis so with him dropping off I was able to control the situation a bit better."
Vettel's Red Bull was by far the quickest car in Melbourne this weekend and recorded the fastest-ever lap around Albert Park of 1:23.529 to take pole position in Saturday's qualifying.
Ferrari's Fernando Alonso missed out on the podium in fourth place, while Vettel's Australian team-mate Mark Webber finished fifth for the third time in his home GP.
"I think we can be very proud of ourselves, the guys did a great job coming into this weekend and a week or two ago we were not expecting to be anywhere near the top five here," Hamilton said.
"So to come away with second, the car was reliable, the longest the car has lasted, and I think it's a great achievement for us and clearly we were catching Sebastian early on in the race."
It was Petrov's first podium finish in 20 GPs.
"I am very happy to be on the podium with these guys and all week was pretty good for us and during practice and qualifying our car looked pretty strong and today the team did everything perfectly," Petrov said.
Vettel safely sped away off the pole and headed off Hamilton and Webber to the notorious first right-hander, which claimed seven-time world champion Michael Schumacher, who was forced to pit before continuing.
Jenson Button was enmeshed in a ferocious dice to get past Ferari's Felipe Massa, which enthralled race-watchers for laps with the McLaren driver unable to slip past the dogged Brazilian.
Button finally got past Massa on lap 12 as Webber became the first of the major contenders to pit to change tyres.
Button incurred a drive-through penalty by race stewards for his manoeuvre to overtake Massa.
Alonso made his first stop on the 13th lap, while Vettel made his first change on the 15th lap, taking just 22 seconds to get back on to the track.
Hamilton pitted on lap 16 and took just 23 seconds to rejoin the race, with Vettel quickly regaining the lead.
Schumacher's race was over on lap 23 as he retired to the Mercedes team garage.
Schumacher's teammate Nico Rosberg made it a grim day for Mercedes when he pulled out one lap later with mechanical problems after coming into contact with Rubens Barrichello's Williams' car.
Barrichello was subsequently given a drive-through penalty for his part in the incident.
Vettel came in for his second tyre change on lap 36 - 22 laps from the finish - with Hamilton also making his second stop as the German's lead grew to 12 seconds.
His teammate Webber pulled in for his third stop on lap 42 temporarily ending his duel with Alonso, who moved to third. Alonso had his third stop on the next lap.
Vettel cruised to the chequered flag underlining Red Bull's dominance in the early stages of the new Formula One season.