Italy started brightly and dominated throughout but after missing several great chances early on Antonio Conte's men fell behind when Alexander Tettey beat Gianluigi Buffon from distance after latching on to a headed clearance.
Italy, unbeaten throughout, stuck to their task and on 73 minutes Roma's Florenzi pounced at the back post to tap the ball past Orjan Nyland, with Graziano Pelle wrapping up the points minutes later with a cheeky touch to Florenzi's superbly-weighted cross that bounced over Nyland at the 'keeper's far post.
The result leaves Italy, beaten finalists at Euro 2012, unbeaten in the group and with a four-point cushion on Croatia, who beat Malta to move up to second place at Norway's expense to claim the second automatic qualifying spot.
Conte was delighted with the win, but still flummoxed by a UEFA rankings system that means Italy will not be seeded for the competition's draw later this year.
“We've finished top of the group after giving it everything against Norway,” Conte told Rai television after the result, which sees Norway drop into the play-offs. “I'm sorry for Norway, but that's football.”
“I'm happy because we were unlucky to go a goal behind and we came back and did our job. We're on the right road. The lads' attitude is far more important than whether we win or not.”
He added: “We've finished with 24 points. It's hard for me to understand why we're not one of the seeded teams.”
After their 3-1 win over Azerbaijan on Saturday, Conte tinkered with his tactical formation, replacing the 4-2-4 that worked wonders in Baku for a 3-5-2 that appeared to give the hosts even more attacking prowess.
Norway came looking for the shock win that would secure qualification but the visitors were ball-chasing from the outset.
But despite Pelle and Roberto Soriano both coming close early on, Norway broke the deadlock on 23 minutes when Tettey volleyed low past Buffon from the edge of the area after a free kick from the left had been headed out into his path.
To their credit, Italy maintained the pressure with Pelle nodding just wide from close in on Riccardo Montolivo's cross while Soriano was close again with a drive that Nyland pushed round the base of the far post.
Nyland had to tip another Pelle header out from under the bar on the stroke of half-time, before denying Florenzi with a great flying save.
Italy resumed in similar fashion after the interval, Pelle just failing to make contact with Florenzi's low cross and then Soriano's strike from outside the area testing Nyland's reflexes.
An equaliser looked simply a matter of time, but an increasingly nervous Conte ordered a number of substitutes to warm up.
When Nyland came out to close down Eder, the diminutive striker was left in pain and soon replaced by Sebastian Giovinco.
For Italy, it was a blessing. Minutes later the Toronto FC striker found space deep on the left side of the area and produced a great chip across goal that Haitam Aleesami attempted to chest back to Nyland.
An alert Florenzi nicked in to fire the loose ball into the back of the net to pull the match level.
Giovinco almost grabbed one of his own minutes later when he pounced on a drive across goal to test Nyland with a vicious strike that the 'keeper pushed two-handed over the crossbar.
Italy should have had their second when Florenzi finished off a great move with Candreva to beat Nyland with a tap-in, although the goal was incorrectly ruled offside.
But in the end Pelle was not to be denied, the big centre-forward using the deftest of touches to send a bouncing first-timer from Florenzi's superb delivery over the outstretched Norway 'keeper in the final minutes.