Daggers boss John Still made no changes following Saturday’s 5-2 win at Solihull Moors, but new signing Sam Ling was named among the substitutes.
The hosts took the lead inside two minutes at Victoria Road when Jordan Maguire-Drew found the net.
The teenager let fly from just inside the area, with his effort clipping in off the post past a stationary Steve Arnold in the Dover goal.
At the other end, Daggers were fortunate not to concede in the 12th minute when Jamie Grimes fired wide after Aswad Thomas headed back across goal.
Corey Whitely went close to a first goal for the hosts with two efforts in the 19th and 21st minute, but both were well saved by Arnold.
The attacker was involved in Daggers’ second six minutes before the break when he was fouled in the box.
Saturday’s hat-trick hero Oliver Hawkins stepped up to take the spot kick, and saw it go in off the right post.
Daggers held their advantage into the break, but almost extended it three minutes into the second half when Hawkins narrowly headed over from 12 yards.
Chances followed for both sides in the remainder of the second half, though none were clear cut as Daggers recorded a first clean sheet in three games.
Daggers: Justham; Widdowson, Doe, Staunton, Raymond; Boucaud, Robinson; Whitely (Howell 87), Maguire-Drew (Ling 86), Okenabirhie; Hawkins
Subs: Cousins, Assombalonga, Benson
Dover: Arnold; Magri, Orlu (Sterling 68), Grimes, Thomas; Parkinson, Moore, Stevenson; Lafayette (Emmanuel 79), Modeste (Jackson Jr. 79)
Subs: Kinnear, Sterling.
Attendance: 1,269 (126 Dover fans)
Referee: Robert Whitton