The Rams edged ahead after 40 minutes when Ben Davies jabbed home a close-range effort following a Craig Forsyth header.
Wes Thomas levelled in the 61st minute when he collected a loose ball, held off a challenge by Richard Keogh, swivelled and swept home the ball from a tight angle for his first goal since arriving on loan from Bournemouth.
Birmingham's second goal from Nathan Redmond was also special. After replacing Chris Burke his pace was always a problem for Derby as he indicated when he cut inside and unleashed a raking long-range left-footed shot past Adam Legzdins.
Clough said: "We played pretty well at times but didn't get a result. We need to do more to get that second goal.
"When you are on top and leading by a goal it isn't going to remain like that in Championship games.
"We should have scored more than one goal in the first half particularly when a team is struggling at home.
"More experienced players would have gone on and killed off the game.
"You couldn't say it was a bad performance. We deserved something out of the game but it was due to our inability to get that vital second goal."