Manager Matt Taylor made 10 changes to the starting line-up from the weekend and gave a first City start to Nigel Atangana, whilst goalkeeper Jonny Maxted came in for the injured Lewis Ward.
Maxted was forced into some early action as well as Dominic Poleon sprung the offside trap to run through on goal – he was denied by the City keeper though who stood up tall and saved well.
At the other end City almost profited from some nervous keeping as Nick Townsend fumbled a cross at the back post gifting a chance for Tom Parkes to send a header narrowly wide.
Then Jayden Richardson got into the box only to see his effort blocked behind for a corner which, when the cross came in, Atangana headed the wrong side of the post.
Alex Fisher then sent another header goalwards, which Townsend tipped over the bar but City got the breakthrough five minutes before the break. It came from a really incisive move in the box that saw Jay play a neat one-two with Atangana before firing low into the bottom corner from 12 yards.
However, Newport still look threatening and Maxted was again called into action in the closing stages of the half to block another one-on-one shot as Abrahams profited on a defensive lapse to get in on goal.
City started the second half brightly with Fisher drawing another good stop out of Townsend with a header from a Jay cross. However, as the half wore on Newport began to come back into the game. Matty Dolan had two good chances to equalise but was denied from eight yards by a good block from Ben Seymour and then from 30 yards by a superb fingertip save from Maxted.
And it was just after that spell of pressure from the hosts that City got the killer second goal as Jay picked possession up on the edge of the box, worked the ball onto his weaker left foot and fired an unstoppable shot into the top corner.
After that the hosts rarely troubled the Grecians goal and City closed the game out to leave them sitting top of the group with six points, needing only a point or better in their final group game against West Ham under-21s to secure a place in the knockout stages.
Newport County AFC: Townsend, Dolan, Abrahams (Hillier, 76), Nurse, Whitely, Maloney (Evans, 76), Poleon, Ekpiteta, Inniss, Jefferies, Watts (George, 45). Subs not used: King, Haynes, Amond, Bennett.
Exeter City: Maxted, Woodman, Atangana (Sweeney, 59), Martin, Tillson, Parkes (Bowman, 59), Jay, Fisher (Chrisene, 78), Richardson, Seymour, Dean. Subs not used: Arthur, Sparkes, Morison, Diabate.
Attendance: 635 (47 away)