Vettel had just kept in touch with early race leader Lewis Hamilton before taking control when the Briton got stuck in traffic following his first pit stop, eventually winning by 9.9s.
It was his first and the team's first victory since the 2015 Singapore Grand Prix.
Vettel picked up only seven podiums on his way to fourth in the 2016 drivers' championship, in a year in which Mercedes won 19 races and Red Bull the other two.
The four-time world champion felt today;s result was a just reward for the Maranello team.
“If you're not part of the team I think it's difficult to realise what this team has done in the last six months,” said the German.
“It's been really tough for us as well, not easy to manage the whole team, but I think the new car in general today is fantastic and a great reward, a big relief for everyone.”
Vettel, however, would not be drawn on questions of his championship prospects, instead preferring to focus on the joy of the moment in Melbourne.
“It's just the tip of the iceberg though, so the foundation has been laid a long time ago and (I'm) really proud to see it coming together.
“But this is one race and I'm sure we'll have a great night, and we did already but we will create some great memories tonight and then take it from there.
“I think we enjoy what we do, the spirit's great in the team, that's why we need to keep it up.”
However, runner-up Hamilton said that the result vindicated Ferrari's pre-season promise which saw them widely tipped as the nearest challenger to reigning world champion Mercedes.
“I don't lie and I am not really good at bluffing, or calling a bluff, or playing poker,” said the Mercedes driver of his pre-race comments regarding Ferrari.
“I didn't really know how quick they were going to be in the race, (but) the pace they showed in testing was obviously true.
“Today their speed particularly in that first stint on ultra-soft tyres was fantastic.
“My real strength wasn't until the right at the end of the race but it was far too late by then.
“But it is great we have a race on our hands.”
Vettel's team-mate Kimi Raikkonen had a relatively uneventful run to fourth position, leaving the Prancing Horse four points clear of Mercedes atop the constructor standings.
Round 2 will be held in Shanghai, China from April 7-9.