The striker linked up with James McAtee to hand us a 32nd minute lead, only for the Eagles to draw level moments later.
It was a quiet opening to the game at Crystal Palace's Academy, with very few chances created by either side.
The enjoyed more of the possession during the early stages, but City's defence held out well to nullify the threat posed by the Eagles.
And that defensive work proved crucial as we hit the front just after the half-hour mark, with McAtee setting up Delap, who finished into the bottom right corner to score his first goal since returning from an injury suffered in the reverse fixture.
However, we would only hold the lead for three minutes, as Palace's No.10 Nya Kirby found the top corner to level the scores.
Into the second half, City's defence were required to ride out a wave of pressure from the hosts, but were able to restrict them to a few off-target shots.
And, just like in the first-half, our defensive efforts paid off as we regained the lead when McAtee and Delap combined again for the latter to slot home his second of the match.
That proved to be the end of the action, as City restricted Palace for the last 20 minutes to secure a crucial three points in the Premier League 2 title race.
HOW WE LINED UP
Mikki van Sas was selected as City's stopper, with Kwaku Oduroh, Jadel Katongo and Tomas Galvez making up the defence alongside CJ Egan-Riley, who captained the side after playing the full 90 minutes in the first team's 0-0 UEFA Champions League draw with Sporting Lisbon.
Crystal Palace Under-23s
Starting XI: Whitworth, D. Boateng, Adaramola, M. Boateng, Quick, Watson, Banks, Wells-Morrison, Gordon, Kirby, Oak-Sakyi
Subs: Steele, Shala, Omilabu, Hannam, Phillips
Manchester City EDS
Starting XI: van Sas, Oduroh, Egan-Riley (C), Katongo, Galvez, Charles, Bobb, Gyabi, Delap, McAtee, Edozie
Subs: O'Reilly, Slicker, Borges, Mebude
WHAT IT MEANS
The result sees the EDS strengthen our grip at the top of the Premier League 2 table, extending our gap over second-place West Ham to ten points with four games left to play, although the Hammers have two games in hand.
Our next fixture will be our penultimate home game of the Premier League 2 campaign, as we take on Liverpool at the Etihad Stadium.
Kick-off is at 18:30 on Sunday 20 March.