The Briton, also chasing a hat-trick of wins at the Monza circuit outside Milan, enjoyed a sunny afternoon with a fastest lap of one minute 22.801 seconds on the supersoft tyres after being second to Rosberg in the morning.
Rosberg's quickest effort then was a 1:22.959, with the German lapping in 1:22.994 after lunch.
"It is work in progress but going in the right direction and I think we will have a good battle tomorrow," Hamilton told reporters, looking forward to another qualifying duel between the dominant pair.
Ferrari were best of the rest in both sessions at their home track with Finland's Kimi Raikkonen third in the morning, but more than a second off Rosberg's pace, while team mate Sebastian Vettel was fourth.
Like the Mercedes pair, their positions were reversed in the afternoon with Vettel using the supersoft tyres to post a lap 0.453 slower than Hamilton.
The Italian team have used the last of their season's engine 'tokens' for a power boost after ceding second place in the championship to Red Bull and Vettel said he was encouraged by the improvement.
"It's a new engine, it's doing what it's supposed to, so that is positive," the German said.
Hamilton leads Rosberg by nine points going into the last European race of the season but the triple world champion can hope to extend that on Sunday.
The last driver to win the Italian Grand Prix three times in a row was the late Argentine Juan Manuel Fangio in the 1950s.
Rosberg, winner in Belgium last weekend, set his best lap of the morning on the soft tyres as did Hamilton, who was one of several drivers to have off-track excursions as they probed the limits.
McLaren's Fernando Alonso provided the only real incident of the first session when he crawled back to the pits with a gearbox problem five minutes from the end.
Dutch teenager Max Verstappen, whose aggressive defending against Raikkonen at Spa last weekend is still the talk of the paddock, was eighth for Red Bull in the morning and fifth in the second session.
"I think today showed that we are still a bit off the podium spot," said his Australian team mate Daniel Ricciardo. "Ferrari has a good buffer on us..."
Verstappen and McLaren's Jenson Button started the day testing a version of the 'halo' head protection device attached to the cockpits of their cars.
(Editing by Ed Osmond)
