Reds suffered a frustrating night at the Ricoh Arena as they went down to their first defeat in six games.
Richie Barker's men mastered the Sky Blues for long spells but were given a lesson in finishing by their hosts and in particular two-goal David McGoldrick.
John Fleck put the hosts ahead early on and McGoldrick doubled the lead against the run of play just before the break.
Billy Clarke thundered home his sixth goal of the season on 56 minutes to give Reds hope but McGoldrick produced a top-class finish nine minutes later and despite more pressure from the visitors they could not turn it around.
The manager made four changes to the side which won at Met Police on Saturday with David Hunt, Claude Davis, Mat Sadler and Mike Jones all coming into the side and Reds started well enough but found themselves a goal down after ten minutes.
Claude Davis's pass was intercepted by Carl Baker who fed McGoldrick on the left. His pass to the corner of the box found John Fleck who drove a low shot which seemed to pick up pace off the turf beyond Paul Jones and into the bottom corner.
The visitors responded positively and for the next 30 minutes they were in control yet the equaliser they deserved eluded them.
Joe Murphy made the first of several saves in the 17th minute when he pushed away a shot from Clarke and with Reds causing plenty of problems to a suspect Sky Blues' defence they went close again when Gary Alexander shot wide when Davis flicked on Mat Sadler's cross.
Murphy made an even better save seven minutes before the break when he blocked a close-range shot from Clarke after Alexander had helped on Dannie Bulman's centre.
Coventry were struggling to get into the Crawley final third but against the run of play they doubled their lead on the stroke of half-time. Davis tried to let the ball run out of play but conceded a corner instead. When it was worked to Fleck he drove the ball into the box and it seemed to take a deflection and fell conveniently for McGoldrick to shoot low past a helpless Jones.
Despite their pressure Reds had fallen down in the most important area - putting the ball in the back of the onion bag.
They put that right ten minutes into the second half when Clarke scored his sixth goal of the season. A lovely piece of skill by Mike Jones enabled him to tee up Josh Simpson for a shot which cannoned off William Edjenguele and into the path of Clarke who drove powerfully past Murphy from just inside the box.
Just as in the first period Reds had the bit between their teeth but once again they were punished when on top and fell 3-1 down after 66 minutes. McGoldrick found a pocket of space 25 yards out and advanced before curling a delightful shot beyond Jones and into the top corner.
Richie Barker made a positive change when he replaced Dannie Bulman with Nicky Ajose and twice Reds came close to getting back into the game. With 16 minutes to go McFadzean made a storming run forward and found Clarke whose angled effort flew wide and then the goalscorer went even closer with a shot that was heading into the bottom corner only to take a deflection off Wood and go behind for a corner.
John Akinde came on for his first taste of league action this season and nearly got on the end of Adams' cross as Reds continued to press, but Coventry held on to leave Reds and the 228 fans frustrated.