The key pass came about midway through the 53-lap race as Hamilton had recovered from a poor start to pull up right behind Rosberg.
Clearly feeling Hamilton's presence behind him, Rosberg braked hard and drove straight through the Monza circuit's first chicane at the end of the main straightaway and was slowed as he slalomed through obstacles on the escape road.
All Hamilton had to do was stay on the track and he easily passed Rosberg and then cruised to his sixth victory of the season and 28th of his Formula One career.
"You have to be prepared for the unexpected," Hamilton said. "I'm quite grateful for today that I didn't lose it, didn't crash into anyone on that first corner. ... I managed to keep my head."
Hamilton cut Rosberg's championship lead from 29 to 22 points with six races remaining.
"Nico was under massive pressure from Lewis and Lewis deserved the win today," said Mercedes executive director Toto Wolff, who revealed a slight smile immediately after the pass. "His pace was awesome and that's why he won it."
Rosberg crossed 3.1 seconds behind, and Felipe Massa of Williams was third, a distant 21.8 seconds back.
"It was a pity it didn't work out today," Rosberg said. "Lewis drove a great race and he deserved it."
It was a disappointing day for the Ferrari drivers before their home "tifosi" -- fans -- as Kimi Raikkonen finished ninth and Fernando Alonso's race ended midway through due to a technical problem that resulted in the Spaniard pulling to a stop at the end of the main straightaway.
Ferrari dropped to fourth in the constructor's standings behind Williams in third. Mercedes holds a massive 182-point lead in the team category ahead of Red Bull.
Hamilton and Rosberg qualified 1-2 and fans had anticipated a battle between the pair at the first corner.
At the Belgian GP two weeks ago, Rosberg finished second after crashing into Hamilton early and sending his teammate out of the incident-packed race -- resulting in disciplinary measures from Mercedes.
Relations between the pair remained icy, as evidenced when Rosberg did not applaud as Hamilton was awarded the trophy.
The throngs of red-clad Ferrari fans who thronged the podium celebration showered Rosberg with boos.
At the start, Hamilton had a technical problem and dropped to fourth heading into the first chicane behind Rosberg, McLaren's Kevin Magnussen and Massa.
Hamilton rebounded by passing Magnussen with Massa's help on the fifth lap then overtook Massa around the outside through the first chicane on lap 10 and began to reduce Rosberg's lead lap after lap.
Meanwhile, Rosberg lost time when he drove straight through the first chicane on lap nine -- a harbinger of things to come.
"I came here with a positive attitude and hoping just for no issues," Hamilton said. "I had another serious one but I managed to pull through it. I'm still looking for one of those weekends where we don't have any of those troubles."
Further down the grid, Marussia's Max Chilton had his race end six laps in when he came into a turn too quick and flew into the barriers.
"I just made a slight mistake and you can't turn the car when it's in the air," Chilton said.
Conditions were ideal, with skies clear and the temperature at 25 degrees Celsius (77 Fahrenheit) inside the royal park that contains the track.
Up next is the Singapore GP in two weeks.