After the game finished 3-3 in extra-time, Blues eventually held their nerve to win 3-2 in the penalty shoot-out following a game overshadowed by controversy.
The flashpoint came in injury time of normal time, as Yeovil defender Byron Webster smashed the ball into an empty net after Birmingham goalkeeper Colin Doyle had kicked the ball out of play following an injury to Blues defender Dan Burn.
Clark was incensed that Yeovil did not give the ball back, and despite the Glovers allowing Lee Novak to equalise unchallenged in extra-time immediately after Luke Ayling put the hosts in front, he said he would be looking to complain about Yeovil's behaviour.
"I might be wrong but my parents made me grow up to be a good sportsman, win lose or draw," Clark said after the game.
"I'm going to go down the right channels to complain. I think something has got to be done. It's always easy to give a goal back when you're leading. We were winning going into injury time.
"I'm proud of my players, every single one of them. I said to them before the penalties that whatever happens, even if they missed or scored, they have done the club proud.
"The way we reacted when that winning penalty went in showed you how dramatic the game was. I've got a group of players who had to play 30 minutes extra and we'll get back to Birmingham late tonight. We're going to be extremely fatigued.
"This club has got great tradition in this competition (having won it in 2011) and I want to carry that on. We want a home draw and one we're capable of winning as I'm excited and this can kickstart our season."
Meanwhile, Yeovil boss Gary Johnson admitted that telling his players to play on was something which he probably should not have done in hindsight.
"I apologised to Lee Clark at half-time in extra-time because, on reflection, it was ungentlemanly. However, we get a bit fed up of teams kicking it out for their own players here when we're trying to get a goal back," said Johnson.
"I wanted us to play on but I didn't expect the Birmingham side to stand still, and didn't expect Byron to hook a goal in.
"I think people should look at this. How many teams are going to kick the ball out for their own players in the last minute?
"It's happened to us for the last three games. I had this gremlin in my head that said play on, but I didn't expect the series of events to follow. I apologise to the Birmingham fans, staff and players, but this kind of thing is happening too often at the minute."