Just 8,162 were present at the Ricoh Arena to see two early goals in as many first-half minutes put the hosts firmly in control.
Freddy Eastwood broke the deadlock in the 15th minute with his third goal in three starts since the turn of the year.
And the Sky Blues looked to have given Palace too much of a mountain to climb just 122 seconds later as Carl Baker found the net with a sublime volley at the back post.
But Neil Danns' sixth goal of the season in the 81st minute set up a frantic finale which saw the Eagles midfielder presented with a golden opportunity to rescue a replay from the spot almost immediately from the restart, but his weak penalty was saved by Westwood.
Danns then missed an even easier chance with his head from two yards as Coventry just about closed out victory to secure their first triumph in seven games.
Palace caretaker boss Dougie Freedman claimed a crucial 1-0 victory over relegation rivals Preston on Monday in what was his first game in charge.
And he went into today's cup tie knowing it may well be his final match at the helm, with Eagles co-chairman Steve Parish yesterday revealing he hopes to appoint a permanent replacement for George Burley next week.
But his hopes of handing over the reins with a 100% record were dashed early on as Coventry raced into a commanding two-goal lead.
Lukas Jutkiewicz had seen his header rebound off the underside of the crossbar 60 seconds before Eastwood opened the scoring, converting Gary McSheffrey's pull back after the winger's great work down the left in getting past his man and to the byline.
Two minutes later and the Wales international turned provider, lofting a deep cross - again from the left - to Baker who was on hand to volley a stunning effort past Julian Speroni.
Palace applied immediate pressure in search of a route back into the tie but Coventry survived the visitors' dominance.
And it was they who threatened - but failed to find - a third goal over the remainder of the first half as they looked to put the encounter well and truly to bed.
Andy Dorman wasted a great chance to set up a potential Palace comeback in the 52nd minute, poking past Westwood but also the wrong side of the keeper's right-hand post.
Palace were the ones who had to force the issue as the tie entered the final 20 minutes and Danns should have handed them a lifeline but he could only head over when well placed.
And the Palace man was at the centre of things over the final nine minutes, stabbing beyond Westwood after he had failed to hold onto his first effort.
But he quickly went from hero to villain as he missed from the spot after Sky Blues debutant Stephen O'Halloran had brought down Wilfried Zaha, before then somehow heading wide from close range.