Ferrari's Sebastian Vettel was runner-up, 16.6 seconds behind, with Force India's Mexican Sergio Perez on the podium for the second time in three races after overtaking Kimi Raikkonen's Ferrari on the last lap.
Rosberg started on pole position, led all the way and also set the fastest lap as he chalked up a fifth win in eight races this season. With 13 rounds remaining, the German has 141 points and Hamilton 117.
"It was a special feeling out there in the car because it felt like I could do whatever I wanted. The thing would just stick to the line, stick to the ground, there was no risk of making mistakes or anything," said Rosberg.
While his cruise in the late afternoon sunshine turned out to be something of a snooze for the global television audience, triple world champion Hamilton provided more entertainment.
The Briton, who had been chasing his third win in a row but finished fifth, sounded increasingly frustrated over the radio as he wrestled with his car's settings without the team being able to help because of a clampdown on 'driver aids'.
The problem eventually resolved itself.
"This is ridiculous guys, I don't know. I'm looking at my dash every five seconds trying to find a switch in the wrong position," Hamilton said over the team radio.
"I might not finish this race as I'm going to try and change everything," he continued. "We don't advise that, Lewis," came the reply from the pitwall.
"Can I make suggestions and you say if it's OK or not?," replied Hamilton. "No, that's not allowed. Let's just get our heads down and focus on the job," he was told.
The Mercedes team's non-executive chairman Niki Lauda told reporters it was a question of engine modes and Rosberg had been able to fix his quicker than his team mate.
However Mercedes later clarified that, saying the situation was more complicated.
Rosberg had made a change that he could reverse whereas Hamilton's had been pre-set before the race and he had no information about where to look.
"All they can tell me is there is a switch error, so I am looking at every single switch thinking, 'Am I being an idiot here? Have I done something wrong?' I hadn't," said the Briton, who said the distraction had been dangerous.
Raikkonen, who had allowed Vettel to pass him after collecting a time penalty for crossing the white lines at the pit lane entry, expressed similar exasperation to Ferrari.
The outbursts were highlights on a day with none of the mayhem predicted after a series of accidents in the GP2 support series, in F1 practice and qualifying.
Nobody crashed, the tight turns around the ancient city walls were safely negotiated by all and the 350km blast down the long main straight produced no drama. All but four cars finished.
"I think people lost a lot of money because they were betting on a safety car. I was expecting a couple too," said Vettel.
Finland's Valtteri Bottas finished sixth for Williams, with Australian Daniel Ricciardo seventh for Red Bull after starting on the front row.
Dutch teammate Max Verstappen was eighth, Germany's Nico Hulkenberg secured a double points finish for Force India in ninth and Brazilian Felipe Massa was 10th for Williams.
