The hosts had to make a late goalkeeping change as Seb Brown replaced Ross Worner, who was held up in traffic caused by an accident.
Michael Smith's header from George Porter's cross put Wimbledon ahead.
But Brown's fumble saw Callum Wilson equalise, while Carl Baker's close-range strike and Callum Kennedy's own goal sealed Coventry's victory.
Play was delayed by 15 minutes to help give first-choice keeper Worner chance to reach the ground, but with the first-choice keeper still not at the ground, Brown had minutes to prepare.
Brown, 23, had yet to feature for the Dons this season, and it was his error that gifted the League One side a way back into the match.
Coventry eventually ran out 3-1 winners and progressed to Sunday's second round draw.