Vettel cracked his first win since Singapore in 2015 after qualifying second and getting the upper hand early on over poleman Lewis Hamilton after the Mercedes driver pitted early.
It was Vettel's fourth win for the House of Maranello.
"Unbelievable! Thank you everyone,” enthused Vettel said on the podium.
"It's what we needed. The guys haven't got much sleep. The car is behaving well. So much fun to drive. A beautiful day. Thank you!"
Ferrari's win validated Hamilton's claim that the Italian squad pre-season speed made the Scuderia the de facto favourite to start the year, but it's early days still warns Vettel.
"It's a long long way ahead. But for now we're over the moon,” Vettel said.
"It's been a hard winter. The start I wasn't entirely happy, a bit too nervous! Some wheel slip. [Hamilton] was better. But I had to take care of Valtteri [Bottas]. I wanted to put the pressure on.
"We're here to fight. Lucky as Lewis came out in traffic. Excellent job. I saw so much support around the track. It was great."
Vettel also insisted on the importance of today's win for Ferrari given its winless streak and difficult 2016 season.
"It's a massive achievement," said the German
"It's been a rough road since last year, but a good one. One that we had to take and lessons that we had to learn.
"Now obviously it peaks, and we win the race. It's all great but it doesn't mean it stops there, it's just the beginning, but it's a nice way to give something back.
"Motivation has been really high but obviously its hemps to wave from the podium. The spirit is very good inside the team and for sure today helped."