Skipper George Glendon scored a stunning goal to win the points after Jorge Intima had earlier equalised David Henen's opener.
The victory lifts Patrick Vieira's men up to third in the Barlcays U21 Premier League table, and with games in hand over their rivals, the stage is set for a rousing season finale.
Windy conditions at Southport hampered City's passing game from the start, with the Toffees' direct style a strong test for Vieira's men.
Both sides struggled to create clear chances, with Callum Connelly firing wide for Everton and Brandon Barker and Olivier Ntcham shooting over for City.
It was Everton who took the lead after 41 minutes when Henen pounced to slot home from inside the area.
City's response was almost instantaneous.
Intima was the scorer, heading in from close range after a City corner to send the sides in level at half time.
City were playing three at the back for the first time this season, with Ashley Smith-Brown, Pablo Maffeo in harness.
The formation helped City win the lion's share of possession, but conditions were a leveller and the game was in the balance until the final moments.
Glendon settled matters, and how.
He latched onto Angelino's cross from the left to volley in a superb winner.
The victory sets City's EDS side up well for Thursday's International Cup game against Celtic.