Striker Zaza, winning just his second cap, opened the scoring with a deflected shot in the 16th minute before defender Bonucci headed home just after the hour mark as the Azzurri marked Antonio Conte's first competitive game as coach with a victory to launch their bid to reach the European Championship finals in France.
It was a functional performance against a home side who were enthusiastic but lacked quality in the final third, and Italy extended their run without defeat in qualifiers to 41 matches stretching back eight years.
"In nine days we have already done great things," Conte, whose reign began with a 2-0 friendly win against the Netherlands last Thursday, told Rai 1 after the game.
"It will not be easy for anyone to come and play in Norway. And I don't remember [goalkeeper Gianluigi] Buffon getting his gloves dirty, while we created numerous chances."
Conte's starting line-up showed six changes to the team that lost 1-0 to Uruguay at the World Cup in June in Cesare Prandelli's last game at the helm, with Sassuolo striker Zaza taking advantage of the absence due to suspension of Mario Balotelli to line up in attack.
And it was the 23-year-old who silenced the packed crowd at the Ullevaal Stadion with the goal that put Italy ahead and broke his scoring duck for his country.
A Mattia De Sciglio ball from the left flank found Zaza, and with the Norwegian defence failing to produce a challenge, he worked his way into the box before sending in a left-foot shot that took a significant deflection off Havard Nordtveit to beat goalkeeper Orjan Nyland at his near post.
Norway, who are attempting to qualify for a major tournament for the first time since Euro 2000, pushed their visitors back for a spell as they sought to draw level, although the closest they came was when captain Per Ciljan Skjelbred fed Joshua King, but the Blackburn Rovers striker flashed a shot over from a tight angle.
Nyland then beat away a fizzing Emanuele Giaccherini strike as Norway restricted Italy's lead to a single goal going into the interval, but a Conte substitution led to the away side doubling their advantage in the 62nd minute.
The former Juventus coach took off right wing-back Matteo Darmian, sending on Manuel Pasqual in his place, and the Fiorentina player needed less than a minute to whip in a glorious cross from the left that was headed home at the back post by the unmarked Bonucci.
There was no way back for Norway after that, and Italy might even have increased their lead late on, but Zaza rattled the bar after being played through by strike partner Ciro Immobile.
"It has been a dream week for me," said Zaza. "Now my objective is to keep my place in the team. The work of a striker is tiring, but if you want to play in this side you need to run."
It was the perfect start to the qualifying campaign for Italy, and one that sets them up nicely for a double-header at home to Azerbaijan and away to Malta next month before a stiffer test in Croatia in November.
Per-Mathias Hogmo's Norway, meanwhile, will hope to bounce back when they face Malta and Bulgaria next month.