Chambers headed Arsenal in front 10 minutes into the second half just moments after he had come on to the pitch as a substitute.
Nketiah scored Arsenal's second 14 minutes later when he capitalised on a weak ball back to Leeds goalkeeper Illan Meslier.
Leeds, which had chances of its own in the first half, was unable to find a response as Mikel Arteta's side saw out the game without further alarm.
Arsenal showed the early intent with Nketiah's shot going just wide while Sead Kolasinac struck the side netting with a low angled attempt.
Leeds weathered the spell of pressure and then had two chances of its own before the half hour; Daniel James tamely shot at Bernd Leno when sent clear before the German produced a superb diving save to keep out Jack Harrison's rasping half-volley.
Leno was the busier of the two goalkeepers in the remainder of the half as he saved a low deflected shot from James, and was relieved to see a Tyler Roberts shot fly wide.
Arsenal was dealt a blow shortly after the interval when Ben White limped off injured, but it soon had reason to celebrate when his replacement Chambers scored with his first touch.
A corner from Smith Rowe was nodded back across by Nicolas Pepe to Chambers, who saw his header blocked by Meslier but goal-line technology deemed the ball had crossed the line.
Nketiah added a second with just over 20 minutes left when he latched on to a backward header from Liam Cooper, nudged the ball over the outrushing Meslier before he shinned it into the empty net.
The goal stifled any hope of a Leeds comeback as Arteta's men saw out the remainder of the game and secured their spot in the last eight.