The crisis club, deducted 10 points on Friday night with CCFC Ltd facing liquidation, fell to Mike Jones' 84th-minute winner having battled back from going two goals down inside the first 22 minutes.
Former Swansea striker Jamie Proctor fired Crawley ahead after only 13 minutes with a rasping shot into the roof of the net.
Defender Joe Walsh doubled the advantage soon after, rising unchallenged at the far post to head in a corner from Nicky Adams
Play was held up for three minutes shortly after when a dozen or so City fans, protesting against off-field events at the troubled outfit, were removed from the pitch.
Directly upon the restart, Crawley striker Gary Alexander headed against the crossbar.
Leon Clarke went close for Coventry shortly after the break when he headed wide at the far post and Callum Wilson's header was also off target.
But Wilson reduced the deficit from close range in the 61st minute after an effort from Franck Moussa was parried by goalkeeper Paul Jones.
Moussa then pounced to make it 2-2 in the 82nd minute following a corner by skipper Carl Baker, but Jones raced on to a throughball from Adams to give the hosts victory six minutes from time.