Hamilton got off the line at the same time as the other cars but immediately fell behind, dropping to sixth before the first corner.
Rosberg, who started second, took the early lead and cruised to his seventh win of the season, second straight this year and first of his career at the storied Monza circuit, which has hosted more Formula One racing than any other track.
"I had an amazing start and that made the win," he said.
Rosberg also cut Hamilton's championship lead to two points with seven races remaining.
Hamilton climbed his way back to finish second.
"Obviously the start wasn't great but it's still a great day for Mercedes Benz and I'm glad to be a part of it," Hamilton said.
Ferrari's Sebastian Vettel and Kimi Raikkonen crossed third and fourth, respectively.
Rosberg celebrated with a series of fist pumps when he got out of his car, playfully jabbing at a TV camera. Then he jumped into a group of his team members who hoisted him into the air.
Already fastest in qualifying, Mercedes' single pit-stop strategy proved better than Ferrari's two-stop plan.
It was the 21st win of Rosberg's career.
After his disastrous start, Hamilton quickly charged up the field, passing Daniel Ricciardo's Red Bull and overtaking Valtteri Bottas' Williams on the main straight.
Hamilton then went ahead of the Ferraris after their second pit stops,
Pushing hard to catch Rosberg in the final laps, Hamilton's wheels locked up at the first chicane and he bounced across the speed bumps, costing him more than a second.
Rosberg's winning margin over Hamilton was 15 seconds, with Vettel 20 seconds behind and Raikkonen 27 seconds back.
Ricciardo finished fifth and Bottas crossed sixth.
Max Verstappen, Sergio Perez, Felipe Massa and Nico Hulkenberg rounded out the top 10.
On a hazy late summer day, red-clad Ferrari fans lined the entire track, with banners dedicated to current and former drivers.
"Michael, All our thoughts for you and your family," read one banner dedicated to seven-time F1 champion Michael Schumacher, who remains in secluded recovery from serious head injuries he suffered in a skiing accident in December 2013.
"You cannot describe the passion, you can only live it," read another banner.
During the singing of the national anthem, fighter jets flew over the main straight, coloring the sky in the green, white and red of the Italian flag.
Soon after the start Jolyon Palmer in a Renault and Felipe Nasr in a Sauber made contact, sending both cars back to the pits with damage. Palmer retired, while Nasr came back out briefly to serve a 10-second penalty for causing the accident.
Daniil Kyvat's Toro Rosso and Pascal Wehrlein's Manor also ran into trouble and retired midway through the race.
Next up is the Singapore GP in two weeks.