Ben Davies was the first player from either side to try his luck, but his effort from outside the box after three minutes went wide.
Davies eventually put the Rams in the lead five minutes before the interval when he found the target from close range.
The goalscorer went close inside the opening 10 minutes of the second half, but was denied by Jack Butland.
Zigic had a decent chance to restore parity for the Blues after the hour mark, but his header from close range went over the crossbar.
Birmingham eventually equalised midway through the second half when Wes Thomas scored from the edge of the Derby 18-yard box.
The comeback was complete 15 minutes from time, Nathan Redmond shooting home from distance to make it 2-1 to the hosts.
Thomas added the icing on the cake in the second minute of injury time with his second goal of the afternoon after combining well with Redmond.