Nketiah opened the scoring against the League One side at Emirates Stadium in the 17th minute, reacting quickest to a loose ball after a combination of goalkeeper Lee Burge and U.S. international Lynden Gooch had blocked Rob Holding's header from a corner.
The Gunners doubled their lead within 10 minutes when Nicolas Pepe, making his first start in any competition since the previous round against Leeds, exchanged passes with Cedric Soares before sweeping home via a deflection.
Four minutes later, though, Sunderland halved the deficit with a goal on the counter-attack. Elliott Embleton played Nathan Broadhead in behind the Arsenal defence and the 23-year-old lifted a clever finish over the advancing Bernd Leno.
Nketiah restored the two goal advantage, clipping in a low ball from Nuno Tavares in the 49th minute and then produced a cheeky backheel finish to make it 4-1 after Pepe nutmegged Dener Hume and delivered a low cross.
"This competition is the place I've been getting starts. I'll do my best whenever I'm asked to play," said Nketiah.
"I'm desperate to play football. All I can do is keep working hard. It'll sort itself out. As long as I play for Arsenal I'll give my all," said Nketiah, who has yet to agree any contract extension.
A positive evening for Arteta's side was completed when 18-year-old substitute Charlie Patino, making his senior debut, converted another low cross from Pepe.
Sunderland, who were relegated from the Premier League in 2017, were cheered on by 5,000 supporters from the North East and manager Lee Johnson said that he was pleased they had spells of good play in the game.
"The bigger picture for us is the learning experience. Now they know the standard. When you feel the quality of the opposition you learn so much more. Arsenal were fantastic," he said.