The Sky Blues looked on course to win in normal time when Harry Maguire scored an own goal 12 minutes into the second half, but David McAllister pulled a late goal back for United to take the tie to penalties.
Maguire missed his spot-kick before Ryan Flynn fired over the bar, allowing John Fleck to send the hosts through with a calmly-taken penalty.
Robins said: "It feels great. You saw at the end what it meant to the supporters.
"In the end, we did it in a good way because they got to see the drama of a penalty shootout and there wasn't that much drama over the course of the 90 minutes.
"We came up against a side that set their stall out to make it difficult to break them down and we found it difficult.
"They came to do their job and did it in a way that made it hard for us to play any expansive stuff.
"I'm not going to complain too much because we have progressed and we got the result we were looking for and we look forward to the next one."