The League One side, competing in the Football League Trophy for the first time in their history, were quick to make an impression on the game with York stopper Michael Ingham producing a fine save at point-blank range to block Richard Wood's header.
Ashley Chambers missed a glorious opening for the Minstermen midway through the first half as he failed to connect properly with Oli Johnson's inviting delivery.
And in the 58th minute David McGoldrick gave Coventry the lead, capitalising on a misplaced pass from Clarke Carlisle to stroke the ball past Ingham.
The Sky Blues, in what was the first meeting between the two sides since 1960, added a second in the 72nd minute as James Bailey dispossessed Scott Kerr in midfield to set free Callum Ball, who casually placed the ball out of Ingham's reach.
Jamal Fyfield unfortunately diverted McGoldrick's effort into his own net ten minutes from time, before Chris Hussey rounded off the second-half goalfest when he ran the length of the pitch to fire beyond Ingham to compile the misery on York.