The televised scenes, with fans ripping up seats and throwing flares after a post-match pitch invasion, came as England wait to hear on Thursday whether they have won a bid to host the 2018 World Cup.
"We wholly condemn the actions of any individuals who have been involved in the scenes of disorder at St Andrews this evening," said a spokesman for the Football Association in a statement.
"The FA abhors acts of violent behaviour and we urge the clubs and authorities to identify any individuals involved in disorder, and impose the stiffest available sanctions, including banning orders.
"The FA will fully investigate the matter in conjunction with the appropriate authorities," he added.
"The whole of English football has worked hard to eradicate scenes of disorder from our game, with Home Office figures this week showing a significant drop in arrests at the many thousands of football matches held every week, the majority of which still pass trouble free."
Birmingham City manager Alex McLeish, whose side beat their city rivals 2-1, was clearly concerned about possible sanctions when he condemned the scenes after the match.
"It takes us back to the Dark Ages," he said of the violence.
