Andrew Nelson headed in an early corner to put Sunderland ahead and Greenwood scored twice in the first 25 minutes of the second half to wrap up victory.
Greenwood was denied a hat-trick by a fingertip save from Charlie Andrew.
The result leaves Sunderland top of the group on six points from three matches, with Brighton & Hove Albion, Exeter City and Hull all on three.
Because Hull are bottom, having played all their matches, they cannot qualify for the next round.
Hull: 22 Jan v Exeter City (A) PL Cup
Sunderland: 10 Dec v Liverpool (H) PL2