The pole sitter, cheered on by Germany World Cup winner Lukas Podolski in the Mercedes pits, extended his lead to 14 points over teammate Lewis Hamilton, who charged up to finish third from 20th on the grid, in the title race.
“It’s an amazing feeling to win at home. It’s a very special day,” said Rosberg.
Splitting the Mercedes men in second was the Williams of Valtteri Bottas.
Rosberg came home a comfortable 20.7 seconds ahead of the Finn at the end of an enthralling 67-laps contest.
Hamilton was only 1.8 seconds behind Bottas ahead of defending champion Sebastian Vettel of Red Bull, Fernando Alonso of Ferrari and Daniel Ricciardo, in the second Red Bull.
Rosberg’s win came after a week in which he had married his long-term girlfriend Vivian Sibold, signed an extended contract with Mercedes and celebrated Germany’s World Cup victory.
It was his fourth win of the season and the seventh of his career.
But even such joyful supremacy was overshadowed in an incident-filled race by the bold passing pace of Hamilton, who started at the rear of the grid following his high speed crash in qualifying.
The Englishman delivered a thrilling series of dramatic moves muscling his way through the field, but in the end was unable to grab second in the closing laps as Bottas hung on to register an historic podium for the resurgent Williams team.
I had great fun,” said Hamilton.
“I did as good as I could. It was hard to get through the pack safely… it was hard to overtake so I’m glad to get some points today.”
On one passing move Hamilton’s car clipped the McLaren belonging to Jenson Button.
“I had a little bit of a collision with Jenson,” he reflected.
“I thought he was going to open the door which he has done a couple of times lately but that was my bad judgement.”
Nico Hulkenberg of Force India finished seventh ahead of Button and his McLaren team-mate, Danish rookie Kevin Magnussen, and Sergio Perez who finished 10th in the second Force India.
After the searing heat of the previous three days, the race started in cooler conditions with an air temperature of 26 degrees Celsius and track equivalent of 33 degrees, compared to the 58 recorded on Friday and 55 on Saturday.
A morning shower played havoc with the team’s tyre-wear calculations in front of an improved attendance after so many empty seats on Saturday.
When the lights went out, Hamilton made a fast start from the back, but the drama was all up ahead.
As pole sitter Rosberg powered through the first bend there was a cloud of dust and debris as Magnussen collided with Felipe Massa, sending the Brazilian’s car into a spectacular somersault.
His car bounced back to a halt on the safety gravel with Massa sitting upright and, amazingly, unhurt.
“I am ok,” he said. “The accident was a little bit more scary watching than being inside. I just saw everything the other way around, but I am fine.
“I am so disappointed at what happened. It’s another race and another car that has pushed me out and finished my race.”
The race resumed two laps later with the circuit cleared.
Hamilton rose steadily through the pack, scrapping his way past all his rivals in a feat of speed and courage that provided thrills at every corner, notably when he negotiated his way past a cluster of cars driven by Ricciardo, Raikkonen and Alonso to reach seventh on lap 13.
By lap 35, despite damaging his front wing after the Button altercation, Hamilton was third behind Rosberg and Bottas. Even damaged, he was the fastest man on track, but Rosberg was cruising at this time with a controlled advantage of 15 seconds with 30 laps to go.
Bottas pitted on lap 41, Hamilton taking second but complaining that his “front left isn’t going to last” before, in a strategy change, following Rosberg into the pits on lap 43 for super-soft tyres.
This set up a final high-speed chase with Bottas managing to hold on in a thrilling climax.