Arsenal, a goal down to a Josh Murphy strike, relied on the 18-year-old forward to first force extra time by netting from close range with his first touch, before heading an extra-time victor for the two-time winners.
The youngster then turned the game on its head six minutes into extra time when he powered in a close-range header from an Elneny left-wing corner to put the Gunners ahead.
Daniel Farke's Norwich took a surprise lead with Murphy latching onto a superb through-ball from Sunday's East Anglian derby match-winner James Maddison before finishing past debutant Matt Macey with a cute dink.
An incisive break in the 55th minute should have resulted in a red card for Elneny, but referee Andrew Madley only gave him a caution for a foul on Oliveira despite the Egyptian being the last man. First, Oliveira blasted over after 69 minutes before Murphy sent a half-volley wildly over with only Macey to beat.More news: Texas town offers hurricane relief, but only with a political promise
More news: Storm Brian set to batter United Kingdom with heavy rain and 70mph winds
More news: ZTE launches Foldable ZTE Axon M with dual touchscreens
With six of his seven subs being inexperienced teenagers, Wenger chose to stick with what he had and Arsenal did begin to see more of the ball in the final third.
Gunn saved superbly from Jack Wilshere in the last minute before Akpom should have won it for Arsenal deep into added time.
Norwich were denied a late penalty when James Husband was hauled down in the area by Mathieu Debuchy, as Arsenal held on thanks to Nketiah's memorable intervention.