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 18-year-old stabbed in the equalizer with his first touch, barely 15 seconds after coming on and scored the victor in extra time with a bullet header.
However, the game would see extra-time, with Nketiah's home debut getting even better, courtesy of the victor on 96 minutes.
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.
However, it was the visitors who were the better side for long periods of the game, with Josh Murphy opening the scoring on 34 minutes, via a deft dink over debutant 'keeper Matt Macey.
Accompanying that embarrassment are losses to Bradford City and Sheffield Wednesday more recently, which is what puts the doubt into Arsenal fans' heads going into this game.
But Farke pointed to an incident after the interval where Elneny escaped a red card after he hauled down Nelson Oliveira as the Portugal global burst through on goal as a key point in the contest.
Walcott headed onto the roof of the net as the final 10 minutes arrived, but Arsenal did draw level with five minutes to play through Nketiah, who stabbed home a flicked corner 30 seconds after being introduced.
Nketiah would go close to completing a memorable hat-trick, only to be thwarted by 'keeper Angus Gunn.
The visitors were all but out on their feet and could not find an equaliser as Arsenal saw the game out to progress to the quarter-finals.
In the wing-back positions Nelson and Maitland-Niles may get the nod to start.
Subs not used: OIliev, Osei-Tutu, Sheaf.
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