The 83rd ranked New Zealanders controlled much of the game this morning (NZ time) and coach Darren Bazeley would have been happy with an improved effort in front of goal.
Bazeley had asked for their chances to be put away and his side responded.
Marko Stamenic had a goal disallowed in the 13th minute.
Stamenic scrambled the ball home when the ball was headed back into the goal area from a free kick, but VAR decided a player was off-side.
The Norwegians, ranked 31st, also had opportunities in the first half, with goalkeeper Alex Paulsen called on to tip a shot over the bar.
But the Kiwis deservedly went ahead just before half time when Finn Surman poked a cross in at the far post from a free kick.
Chris Wood then had a couple of chances after the break, including hitting the crossbar.
The Nottingham Forest striker was subbed off in the 62nd minute and the home side then got on level terms when Antonio Nusa curled a strike past Paulsen.
Norway, who were playing without Manchester City star Erling Haaland, then had a period of domination as the game ebbed and flowed, but the All Whites defended well while Paulsen continued to impress.
Both sides scrapped for the winner in the final quarter - Ben Old having the best chance for the All Whites with his header saved on the line by the keeper.
While it was an improved effort, missed opportunities in front of goal did cost the All Whites a chance of a top-ranked scalp.
Bazelely had made just two changes from the side that lost 1-0 to Poland last week with Wood starting up front and Old replacing Callum McGowatt on the wing.
It was Wood's 88th international, joining Ivan Vicelich as the most capped All Whites.
New Zealand play Colombia and Ecuador in the next FIFA international window in November, as they continue their preparations for next year's World Cup in North America.
