Several hundred home fans ran on to the St Andrew's pitch after the 2-1 win, goading the Villa fans in the Gil Merrick stand and precipitating further trouble between the two groups.
Even more worryingly, flares were fired across the police divide between rival fans.
Flame thrower: a flare is hurled as Birmingham City's Carling Cup win is marred by disturbances after the final whistle
McLeish said the victory celebrations were over-thetop and that, had the roles been reversed at Villa Park, the same would not have happened.
'We have to be bigger than that,' he said.
'We ought not to have a small-club mentality. The scenes were like when Hereford beat Newcastle in the FA Cup and fans ran on at the end. That's fine. But not at Birmingham, not a Premier League club. Our fans don't need to do that.
Making a scene: more than a thousand fans stormed on to the pitch
'Definitely celebrate it, but don't make it look as if it's the World Cup final.
'It's a kind of phenomenon for us that we have beaten them. It's a huge thing for our fans. I don't condone them running on at all but I understand their emotions.
'Running on the pitch is just a no-go in modern-day football.'
McLeish insisted the trouble had not affected the outcome of England's failed 2018 World Cup bid.
'It's ridiculous,' he said.
No go zone: McLeish says fans should not run on to the pitch in the modern game
'I think it was decreed before that game the other night.
'The vibes leading up to Thursday's decision had everybody a little bit worried, regardless of what happened at the Villa game.'
The Birmingham manager also denied that the stand-offs occurred because the derby was in midweek.
'If it was a Saturday the fans would probably have still run on, such were the emotions of beating Aston Villa.
'Villa fans are used to beating Birmingham so they wouldn't have run on if they had won, even at their home ground.'
Birmingham promise to ban hooligan fans after Carling Cup violenceFA and police probe violence in derby between Birmingham and Aston VillaBirmingham derby marred by violence as Blues dump rivals Villa out of Carling Cup Birmingham 2 Villa 1: Towering Zigic is up for the cup, firing hosts into last four Explore more:People: Alex McLeish Places: Birmingham, Hereford, Newcastle, United Kingdom