City responded well to the setback and looked certain to equalise six minutes later when Luke Boden passed across the face of goal for Cyle Larin, but his side-footed effort was straight at goalkeeper Steve Clark.
However, the hosts did eventually draw level on 18 minutes when Larin squeezed between two defenders to tap home Corey Ashe's perfectly-weighted ball.
It proved a crucial intervention as Darwin Ceren completed the turnaround on 38 minutes when his vicious half-volley from outside the six-yard box flew into the top corner for a half-time lead.
In the second half, Larin extended his side's advantage when he latched on to Ashe's low cross into the box before striking a first-time left footed effort past Clarke into the bottom corner.
Aurelien Collin put the game beyond doubt for the home side when he powered a header past Clark from Boden's corner at the near post.
City added a fifth goal when Cristian Higuita guided the ball past Clark from close range with his head, although the Crew got one back just before the full-time whistle as Collin steered the ball into his own net.