Although Mercedes drivers Valteri Bottas and Lewis Hamilton started with a strong showing, Hamilton's run was hobbled by a false start.
Bottas claimed his maiden F1 victory on Sunday while Hamilton was forced to settle for fourth behind the two Ferraris. Red Bull is a distant third with 57 points.
"A big congratulations to Valtteri, he did an exceptional job all weekend - he was quickest".
"Now I have my first, hopefully it is the first of many in F1 now that I know how to do it - I think today I proved I can win to the team and also myself".
Lewis Hamilton shrugged off a disappointing Russian Grand Prix qualifying on Saturday, with Ferrari sweeping the front row for the first time in almost nine years, as "just one of those weekends".
And retired three-time title victor Lauda believes Bottas can now push on in search of further wins, having taken his first chequered flag in just his fourth race for Mercedes.
"This unusual opportunity came over the winter to join this team and I have to thank the team for making this possible", he added after meeting Russian president Vladimir Putin before spraying champagne on the podium. Tell us in the comment section below.
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"In the race I had overheating, but I think I could have kept up with the guys ahead". His win shows it's not clear-cut who's better suited to take the fight to Ferrari.
Hamilton congratulated the Finn after the finish, and Rosberg sent a congratulatory message on Twitter, before Bottas took a hefty swig of the podium champagne and breathed a sigh of satisfaction and relief. Bottas meanwhile moves up to 63 points, 10 behind Hamilton, and 23 behind Vettel - the closest the championship top three has been after four races in five years.
Fifth place went to Max Verstappen, followed by Sergio Perez in sixth and Esteban Ocon in seventh.
Bottas went for the pit-stop on Lap 28 giving the lead to Vettel, who lingered on the track a while longer going for his pit-stop about ten laps later.
There was a safety-car stoppage after barely half a lap when Romain Grosjean's Haas hit the Renault of Jolyon Palmer on the inside of a right-hand bend, putting both cars into the wall.
Red Bull's Daniel Ricciardo was the most notable retirement, parking his auto on lap 6 after the right-rear brakes caught fire.
Mercedes chief Toto Wolff expects an intense fight between Bottas and Hamilton this season, but he insists the dynamic is very different to the one the team has endured recently.