Needing a win to finish second in Group H and clinch their place, Norway were leading through Alexander Tettey's first-half goal until a blunder by Haitam Aleesami allowed Alessandro Florenzi to level.
Graziano Pelle then snatched a late winner to ensure top spot for Italy while Norway finished third, below Croatia who beat Malta 1-0, and were left facing a two-legged playoff.
Helped by some fine saves by Orjan Nyland, Norway survived a lively Italy opening and took the lead with their first attack when Tettey blasted in a half volley from the edge of the area after the hosts only partially cleared a free kick.
Nyland continued to defy Italy as he turned Roberto Soriano's effort around the foot of the post, tipped away Pelle's header and then made a superb stop from Florenzi before halftime.
Norway tried to pack their defence and waste time after the break and their tactics frustrated Italy, who appeared to be heading for their first competitive home defeat since they lost to Denmark in September 1999.
But it went wrong for the visitors when Aleesami tried to chest the ball back to Nyland but apparently failed to notice Florenzi, who prodded the ball into the net in the 73rd minute.
Florenzi also set up the winner when his cross found Pelle who took his time before guiding the ball into the net with eight minutes left.
(Writing by Brian Homewood in Berne; editing by Toby Davis)
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