Green's first and a penalty from Callum O'Hare put Villa 2-0 up at the break, before Green secured the points with a fine strike from an acute angle.
Villa top the group on nine points, three ahead of Ipswich, who have a match in hand, and Sheffield United, after the Blades won 2-0 in Monday's meeting between the two sides.
The Baggies' third successive group defeat leaves them bottom of the table and looking for their first point.
West Brom: 25 Jan v Ipswich (H) PL Cup
Villa: 29 Jan v Middlesbrough (H) PL2