They doubled their lead in the 18th minute when McEvilly cross was missed by Justin Whittle, and Darren Hockenhull finished.
City were totally under the cosh, and Alan Fettis was alert to stop them extending the scoreline.
The second half was a different story though as City got back into the game.
Ben Burgess pulled a goal back in the 64th minute, heading home Dean Keates' free kick.
Matthew Gilks was now the busier of the two goalkeepers, but Dale held on for the 3 points with a 2-1 win.
City line-up (4-4-2): Fettis; Otsemobor, Whittle, Anderson, Delaney; Regan (Reeves 55), Melton (Burton 59), Keates, Elliott; Walters, Burgess (Webb 84).
Un-used subs: Musselwhite, Peat.