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Derby had led from the 41st minute thanks to Rob Hulse, but Morrison levelled after bundling an effort past keeper Stephen Bywater in a crowded penalty area.
The game started well and Derby could have been a goal up in the eighth minute after good work by Nathan Ellington.
He dribbled through the Coventry defence only for his low shot to be stopped by goalkeeper Keiren Westwood's feet.
Coventry's first decent attempt came ten minutes later when Freddy Eastwood beat two Derby defenders and fired an effort towards the bottom corner from 20 yards only for Bywater to tip around the post.
The tempo had slowed but a sudden burst of pace from Jordan Stewart in the 41st minute helped lead to the first goal.
Stewart laid the ball to Kris Commons on the left and his cross was met by Hulse with the ball looping over Westwood and into the net.
The goal sparked Coventry into life and they could have been level two minutes later.
Scott Dann's through ball found Aron Gunnarsson eight yards from goal but the Iceland international's shot hit the post.
Coventry came out fighting in the second half and Gunnarsson had another chance to level but his shot from 18 yards deflected into the arms of Bywater.
Eastwood also tried his luck and fired a shot from 25 yards but his effort flew just past the bar.
Derby were happy to sit back and soak up the pressure which forced Chris Coleman to change the dynamic of the game.
He introduced Morrison as a fourth striker and the home side again started to pressure.
Gunnarsson hit a half-volley from 16 yards which Bywater saved well, before Stewart came up with a last-gasp clearance as the City strikers waited to hit the ball home.
But just as Derby seemed to be leaving with three points, Michael Mifsud slid a pass across the crowded box and Morrison scored his first league goal for Coventry.