The German was handed his winner's trophy by Russia's president Vladimir Putin, who was spotted enjoying the race alongside Formula One chief executive Bernie Ecclestone.
Putin was on hand to present the trophy to the winner in 2014 and 2015, when he was infamously caught in the champagne crossfire as Lewis Hamilton celebrated his victory.
Russia's Olympic city is set to host the Grand Prix for another four years, after securing the rights to the annual race from 2014-2020.
Hamilton, Rosberg's Mercedes teammate, finished more than 25 seconds behind Rosberg in second place, after starting the race in 10th following his early withdrawal from the third qualifying round Sunday.
Finnish driver Kimi Raikkonen finished in third place.
"It's been an awesome weekend -- the car's been fantastic," Rosberg told reporters after the race. "I'm very, very happy, thanks to everybody. If you slow down, you make mistakes.
"Great job from Lewis (Hamilton) to come up to second so quickly."
Rosberg's victory was his seventh consecutive win -- taking into account the final three races of last season -- and sees him extend his lead over Hamilton at the top of the drivers' standings to 43 points.
"It wasn't the easiest first corner," Hamilton said after the race.
"I'm really happy for the team. There wasn't a doubt I could win it, but then I had the problem with the engine and had to back off."
Four-time Formula One world champion Sebastian Vettel was forced to retire just two corners into the race after twice being hit by Daniil Kvyat, prompting Vettel to launch an expletive-ridden rant down his Ferrari team radio.
"I got a big hit and was lucky not to spin into turn two," Vettel told Sky Sports after his retirement. "Then another huge hit which turned me around and I couldn't avoid hitting the barriers.
"There's nothing I could have done differently. If anybody needs to talk to anybody, I think it's him (Kvyat)."