With a minute of normal time remaining City looked to be going out of the competition courtesy of the 1-0 home defeat earlier in the month.
At this stage their dismal display deserved nothing less but a moment of good fortune saw defender Danny Califf score an own goal in the 89th minute to force extra-time and extended their interest in the competition.
Boss Mark Hughes admitted his side has been under par but the result was all that mattered.
"It was difficult for us but we've got through so it has been a positive day all round," he said.
"We kept on knocking on the door. They were in the box-seat and could afford to get people behind the ball and play the counter-attack.
"We always had to be mindful of that but, in the end, we got our rewards.
"After an hour's play I thought we upped our energy levels and our play was more progressive.
"From that point onwards, through extra-time, there was only one team that looked like they were going to nick it.
"We had a couple of chances, Michael Johnson could very well have won it before the end of play.
"But we'll take the result, it doesn't matter how we go through.
"It's a bit of a lottery with penalties but we're delighted we've overcome this hurdle."
Hughes added: "You're always relieved. There are no easy games; it's a cliche but in Europe there aren't. Look at the events of last night with Liverpool.
"I'm delighted, we showed a lot of character and I saw qualities and values in the team that I really liked tonight."