The 31-year-old German took the lead at the start, when pole-sitter Hamilton became bogged down, and pulled away to control the race and finish 15 seconds clear of the defending three-time champion.
Hamilton, who effectively lost the race in the first 20 metres when the lights went out, now leads the title battle with 250 points ahead of Rosberg on 248.
Sebastian Vettel came home third, 5.9 seconds further adrift, ahead of his Ferrari team-mate Kimi Raikkonen to the delight of the home fans.
"Thank you very much guys," said Rosberg. "It's great to win Italy."
It was also the 50th podium finish of Rosberg's career and prevented Hamilton completing a cherished hat-trick of Italian wins, to equal a feat achieved only once before by Juan Manuel Fangio in the 1950's, and register his 50th career victory.
It was the first time in seven years that the race was not won by the driver starting from pole position.
"Good job!" grimaced Hamilton as he shook hands with Rosberg afterwards.
After three days of searing late-summer heat, conditions were slightly cooler when the lights went out with an air temperature of 29 degrees and the track at 38 in the old royal park, some 20 kms north of Milan.
Hamilton's initial reaction saw him depart cleanly only to lose momentum as Vettel, from third, passed him and Rosberg took the lead. The champion, as if glued to the asphalt, fell to sixth.
For Hamilton, running on soft tyres, the race became a strategy exercise. "Remember, Lewis, the three cars ahead are on super-soft tyres so they'll be quick for a few laps and then they will degrade," he was told.
At the front, Rosberg was pulling clear with a run of fastest laps while Hamilton scrapped to pass Bottas for fourth. After 10 laps, Rosberg led Vettel by 4.4 and Hamilton, who eventually surged past the Finn on the main straight, by 11 seconds.
The Ferraris were soon in, both for more super-softs, indicating that the scuderia were adopting a two-stop strategy while Mercedes, with Hamilton inheriting second place 15 seconds behind Rosberg, sticking to 'Plan A'.
By lap 20, Hamilton had cut the gap to 13.4s, reporting that his "tyres are fine" as he continued his pursuit of the German.
After 24 laps, Rosberg pitted from the lead for mediums. A 'slow' 3.9s stop saw him emerge second behind Hamilton - leading for the first time - until the Briton pitted and rejoined behind the two Ferraris.
Vettel pitted again after 33 of the 53 laps, rejoining fifth ahead of a rampant Verstappen. A lap later, Raikkonen did the same and also took softs, slotting in behind the Dutchman.
With 17 laps remaining, Rosberg led Hamilton by 11.5 with Ricciardo third, until he took his delayed second stop for more super-softs to equip him for a final high-speed scrap with Bottas.
The F1 circuis moves next to Singapore.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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