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.
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.
It was nearly 2-0 on 39 minutes when Murphy picked out Nelson Oliveira on the edge of the area and his curling effort towards the far corner was pushed away by Macey.
Arsenal started the second half by committing plenty of men forward and were nearly caught out by a simple through ball which set Oliveira away.
Arsene Wenger will rotate his Arsenal side once again for their Carabao Cup fourth-round clash with Norwich tonight.
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Meanwhile, monthly input price inflation eased notably to 0.4 percent from 2.3 percent. Inflation was last higher in March 2012.
Arsenal were fortunate not to be down to ten men on 58 minutes when Mohamed Elneny appeared to be the last man when he brought down Oliveira but was only shown a yellow card.
Murphy then wasted a half-chance before Mario Vrancic should have sealed Norwich's spot in the last eight, only to shoot wide after working space in front of goal.
Another opening came and went for Walcott early on in extra time, but Nketiah had less trouble with his finishing as he nodded the ball past Gunn from a left-sided corner.
Norwich then missed a hat-trick of opportunities to put the game out of sight.
The visitors were all-but out on their feet and could not find an equaliser as Arsenal saw the game out to progress.