The flashpoint came in injury time of normal time as Yeovil's Byron Webster lobbed 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 - with City boss Lee Clark furious the ball had not been given back.
Johnson admitted he told his team to play on and not return the ball from the throw-in having grown frustrated this season by what he perceives as time-wasting tactics.
Blues eventually held their nerve to win 3-2 in a penalty shoot-out after the match finished 3-3 following extra-time.
Johnson said: "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.
"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."
With the scores level at 2-2 after 90 minutes, Yeovil moved ahead in extra time through Luke Ayling, but then immediately allowed the visitors to walk in an equaliser from Lee Novak as compensation for Webster's controversial effort.
Tom Adeyemi's penalty in the shoot-out took the 2011 League Cup winners into the third round but Clark still accused Yeovil of poor sportsmanship and vowed to explore every avenue to complain about the Somerset side's conduct.
"I might be wrong but my parents made me grow up to be a good sportsman - win, lose or draw," he said.
"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. We're going to be extremely fatigued."