Former Sheffield United striker Clarke had a goal-bound shot parried by goalkeeper Joe Murphy early on before firing Crawley ahead in the 15th minute.
A free-kick on the edge of the penalty area by Nicky Adams was worked to Clarke and his deflected low shot wrong-footed Murphy.
The hosts threatened again midway through the first half when a dangerous cross by Clarke was headed over by Mat Sadler.
Coventry skipper Carl Baker forced Crawley goalkeeper Paul Jones to parry his free-kick and came close again moments before the break when Jones palmed away his goalbound shot.
The match was played in constant rain and it was difficult for both sides on a heavy pitch at the Broadfield Stadium.
But former Gillingham defender Essam sealed victory in the 59th minute with his first goal for the club, bundling the ball home from close range following a corner by Adams.
Coventry never threatened to launch a comeback as they slid to their third successive away defeat.