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.
Nketiah, 18, then equalised with his first professional goal just 15 seconds after coming off the bench.
Wenger made 11 changes from the team that won 5-2 at Everton on Sunday and they fell behind in the first-half as they struggled to gel.
Wenger named Olivier Giroud, Theo Walcott, Jack Wilshere and Alex Iwobi in his starting line-up, but none of them could make an impression against a stubborn Norwich defence, which only wilted following the introduction of Nketiah.
Norwich took the lead in the 37th minute after Josh Murphy raced onto a through pass before chipping Arsenal keeper Matt Macey. Wilshere and Coquelin will likely start in the center of the park for Arsenal.More news: James Comey confirms he is Reinhold Niebuhr on Twitter
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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.
"There is no pressure on because 19 times out of 20 Arsenal would win", he said.
He entered the fray in the 85th minute to taunts of "Who are You?" from Norwich supporters, but Edward Nketiah left to chants of "Eddie, Eddie" after making a name for himself with two goals in the first ten minutes of his domestic debut to turn this game around and spare Arsene Wenger's blushes.
A minute later the Gunners nearly went ahead when Rob Holding got up to meet a left-wing corner but his powerful close-range header from a Theo Walcott cross was brilliantly pushed over the crossbar by Angus Gunn.