A first half goal from Ben Knight gave the City youngsters an early lead, but we were pegged back five minutes into the second half as Thomas Cannon equalised for the visitors.
It was Maresca's men who proved the more dominant in the opening stages, and this control was nearly rewarded through Knight 10 minutes in.
The City number-nine drove at the Everton backline before dragging his effort from the edge of the box just wide of the far post.
In spite of the miss, the 18-year-old did not have to wait long for his second goal in as many games as he opened the scoring just minutes later.
A smart pass into the box found Knight, and he made no mistake this time as he smashed his effort past Everton keeper, Nicholas Defreitas-Hansen on the angle.
The goal boosted City's confidence going forward and captain, Tommy Doyle tried his luck from range twice in the space of a minute, drawing good saves from Defreitas-Hansen on both occasions.
Callum Doyle was the next to come close for Maresca's side, as the defender latched onto a cross from his captain and crashed a header against the crossbar from 10-yards.
For all of City's first-half dominance, Everton got themselves back into the game five minutes into the second half, as Cannon levelled things up after heading home a corner.
The setback did not deter Maresca's men though, as Palmer put City back in front just minutes later.
The Wythenshawe winger dropped his shoulder to beat his man before unleashing a right-foot effort, which deflected off Mackenzie Hunt into the top corner for his fifth league goal of the season.
Soon after, City had a third as a Edozie finished off a superb solo run by smashing home into the bottom corner to further extend our lead and seal three points.
HOW LINED UP
WHAT IT MEANS
The win means Maresca's side go top of the Premier League 2 table on 25 points after 13 games.
Our EDS return to action on Monday 15 February, as we make the trip to Leicester City in Premier League 2 action.