Chris Coleman's men looked set to ease through to the second round when they took the lead just after the half-hour mark.
But the visitors drew level just before the break and made City sweat until a brace from Robbie Simpson edged Coventry into the next round.
Aldershot had the first genuine effort on goal after only four minutes as John Grant raced clear of Coventry's offside trap.
He looped the ball up and over City's debutant keeper Danny Ireland but Ben Turner headed off the line to save the Championship side.
The home team started to get into their stride and midfielder Guillaume Beuzelin's eighth minute shot produced a fine save from Nikki Bull.
Freddy Eastwood, who was looking much more lively than he did on Saturday during his debut, fired two effort on 11 and 13 minutes, the second of which produced a parried save from Bull.
Aldershot were playing some neat football, but it was Coventry and particularly Eastwood who looked the most threatening.
The former Wolves striker cracked a fierce shot from 22 yards after 20 minutes which forced another strong save from Bull.
Six minutes later, Eastwood turned provider and, after beating Anthony Charles, he crossed to the back post for Simpson but his header was turned away by the impressive Bull.
Coventry's pressure finally told in the 31st minute and Eastwood again was at the heart of the move.
His run and inch-perfect pass put the ball on a plate for Clinton Morrison, who opened the scoring on the night and his account for his new club.
The visitors hit back six minutes later but their equaliser owed a great deal to good fortune.
Scott Donnelly's free-kick from 25 yards took a deflection off his own player which wrong-footed Ireland and bounced into the Coventry net.
The Sky Blues came close to retaking the lead seven minutes into the second half but Eastwood blasted over the top from 22 yards.
Coventry edged ahead after 58 minutes when skipper Scott Dann did well to win back the ball after a Danny Fox corner.
It landed at the feet of Simpson, who buried the ball past Bull at the near post.
Michael Mifsud twice went close for City before Simpson doubled his personal tally by firing into the bottom corner after great play from Eastwood in the 65th minute.
Eastwood had two golden chances to open his account for the Sky Blues but missed the post by inches on both occasions.
For Ian Morsman's match day photo's, click on the link below:
http://www.theshots.co.uk/cclc08Gallery.ink