There was drama before kickoff, as Wimbledon goalkeeper Ross Worner had to be replaced in the starting lineup by Seb Brown after getting stuck in traffic.
The home side had the better of the chances in the first half, with Harry Pell and Michael Smith both testing the Coventry keeper, but the sides headed into half time goalless.
The League Two outfit broke the deadlock early in the second half, as Smith headed in from a George Porter cross.
However, the visitors were back on level terms just three minutes later, will Callum Wilson tapping in from close range after Brown spilled a Cyrus Christie cross.
Carl Baker put the Sky Blues in front in the 59th minute, flicking the ball past Brown after being set up by Franck Moussa.
Coventry sealed the win 10 minutes later, as a Baker free kick was deflected in by Callum Kennedy.
Smith and Sammy Moore both had chances to get the hosts back into the match, but Joe Murphy remained alert in goal to keep their efforts out, securing his team a place in the next round of the tournament.