Despite declining a new deal at Emirates Stadium and being out of contract in the summer, the back-up striker shone in a 5-1 win in which Charlie Patino made his first-team debut.
In the first half, Nketiah opened the scoring before Nicolas Pepe doubled it, while Nathan Broadhead scored his sixth goal in six games to pull one back for the Black Cats before the break.
Any threat of a shock was extinguished within 13 minutes of the second half as Nketiah provided two smart finishes and Patino came off the bench to add the gloss with a late fifth.
Mikel Arteta’s side sit fourth in the Premier League and showed nine changes from Saturday’s win at Leeds as only Ben White and Martin Odegaard kept their places.
Sunderland wanted an early penalty as Ross Stewart went down in the box under close attention from Mohamed Elneny – but there appeared to be no contact and the claims were waved away.
A stroke of luck almost put the hosts ahead as a Nuno Tavares cross was turned against his own crossbar by Eddie Embleton with Arsenal starting to knock at the door.
The deadlock would be broken soon after as Rob Holding’s header was well-saved by Lee Burge, only for Nketiah to knee home the rebound from close range.
Pepe, the club-record signing given a rare start by Arteta, doubled the lead as he exchanged passes with Cedric Soares before seeing his strike deflected past the helpless Burge.
But if Arsenal thought they were coasting into the semi-finals, they were wrong.
Sunderland would hit them on the break, Broadhead coolly finishing a quick-fire counter from the visitors to half the arrears.
He dismissed the comment as “dad banter” on the eve of the tie but Arsenal showed – in the minutes following half-time – that even their much-changed side were more than enough to see off Johnson’s spirited visitors.
Two flicked finishes from Nketiah would settle the contest, the first from a Tavares cross and the second after some superb work from Pepe – who impressed on his return.
Embleton had hit the post in between but Sunderland were out of the tie as their vociferous following sang throughout.
Arteta was able to hand a first-team debut to highly-rated midfielder Patino for the final 10 minutes, the England Under-19 international replacing fellow academy graduate Emile Smith Rowe and sliding in to convert another Pepe assist to put the cherry on top of the cake for Arsenal.