Crewe hit the woodwork three times and always looked lively, though both sides were wasteful in front of goal.
Coventry started brightly and Andrew Whing forced a great save from Clayton Ince after just four minutes, before Mo Konjic met a Dele Adebola flick and only a Ben Rix goal-line clearance kept the ball out.
Crewe responded well and Kenny Lunt rattled the woodwork twice with two 25-yard free kicks at each end of the half.
Gary McAllister tried his luck from similar range in the 22nd minute, but only hit the back stanchion.
Five minutes later Graham Barrett continued the home side's onslaught as he cut into the area but his left-footed effort hit the sidenetting.
Konjic was lucky on 38 minutes when his weak header on the halfway line cannoned off Steve Jones.
The striker scampered off with Konjic stranded and found Dean Ashton with a well-placed pass into space.
However, with just the keeper to beat, the highly-rated youngster only managed to lift his shot into Coventry's West Terrace.
The game turned at half time when Julian Joachim was introduced for Dele Adebola.
The former Aston Villa man was frozen out by McAllister during the summer, but was right back in favour after just two minutes on the pitch when his inch-perfect cross found his boss at the back post and left him with a simple header to put Coventry in front.
Crewe tried to respond and Jones hit the woodwork once more for the visitors after he caught Richard Shaw napping.
But a Youssef Safri free-kick on 72 minutes found in-form striker Morrell, who headed home from ten yards for his fifth goal in as many league games.
Coventry: Shearer, Whing, Konjic, Shaw, Staunton, Barrett, Safri, McAllister, Jorgensen, Adebola, Morrell. Subs: Arphexad, Davenport, Joachim, Warnock, Mansouri.
Crewe: Ince, Wright, Walker, Foster, Vaughan, Lunt, Sorvel, Cochrane, Rix, Jones, Ashton. Subs: Bankole, Bell, Tonkin, McCready, Edwards.
Referee: J Ross (Essex)