Max Clayton's first-half strike gave Crewe a first home victory in npower League One and left mangerless Coventry still looking for their first league win of the season.
Clayton should have put Crewe in front in the third minute but the striker missed the ball with the goal gaping.
Coventry's Cody Mcdonald headed Carl Baker's cross straight at Alan Martin before Baker turned and shot straight at the Crewe goalkeeper.
Baker blew the visitor's best opportunity when Kelvin Mellor's poor clearance fell to the midfielder six yards out. Baker had all the time in the world to pick his spot but somehow headed over the crossbar.
Alex made City pay with 24 minutes gone. Leitch-Smith mis-cued his shot from Harry Bunn's cross but the ball fell kindly to Clayton who could not miss from five yards out.
Crewe nearly extended their lead soon afterwards but Joe Murphy kept out Harry Davis' far-post header with a superb save.
The Sky Blues came out fighting in the second half but more poor finishing cost them when McDonald cracked Baker's cross onto the crossbar and Stephen Elliott screwed the rebound wide.
Debutant David McGoldrick went close to levelling with a well-struck volley from 25-yards which flew inches over the bar.