The Serbia striker scored with a deflected shot to give the Blues their first win over Villa in nine meetings and a place among the final four teams since they lost the 2001 final to Liverpool.
But the match was marred by a pitch invasion at the end, with fans lighting flares and ripping up seats in scenes reminiscent of the violence that affected English football in the 1970s and '80s - just a day before FIFA's executive committee votes on whether to award England the right to host the 2018 World Cup.
"Fans shouldn't come on to the pitch at any time. It soured it a bit for us," Birmingham manager Alex McLeish said. "I don't think that will affect the World Cup bid. I'm sure that's already decided, and let's hope England gets it because it deserves it.
"It doesn't look good, though, when you see fans running on the pitch like that and carrying on. It takes us back to the Dark Ages."
Police issued a statement referring to the violence as "sporadic outbreaks of disorder at the Birmingham derby."
Some matches between local or historic rivals in England can still attract fans intent on disorder.
But the number of arrests by police at all domestic and international football matches in England and Wales fell 10 percent last season.
Birmingham took the lead in the 10th minute when Sebastian Larsson scored a penalty awarded for Richard Dunne's foul on Lee Bowyer.
The home side almost led 2-0 in the 28th, but Zigic's low shot was ruled out because Liam Ridgewell was offside, and Villa equalised moments later through Gabriel Agbonlahor's low shot to the far post across goalkeeper Ben Foster.
Villa winger Ashley Young had a shot cleared off the line before Zigic's shot from Cameron Jerome's cross deflected off defender Luke Young and looped over goalkeeper Brad Friedel.
Birmingham will play West Ham in the next round after the Hammers beat defending champion Manchester United 4-0 in one of Tuesday's two semifinals.
Ipswich beat West Bromwich Albion 1-0 through Grant Leadbitter's 69th-minute penalty to become the only second-tier side in the semifinals, where it will play Arsenal - a 2-0 winner over Wigan on Tuesday.
"In terms of the draw, it is the hardest we could have got," Ipswich manager Roy Keane said. "But it will be great for the players."
