Curtis Davies headed Blues ahead before Burnley moved into the ascendancy.
Martin Paterson had three early chances while leading marksman Charlie Austin headed home what would have been the equaliser only for referee Eddie Ilderton to blow after Kieran Trippier - who crossed for Austin - was fouled by Mitch Hancox.
A Danny Ings strike and Ross Wallace's spectacular free-kick eventually turned the contest on its head after half-time but Zigic had the final word.
Dyche said: "It the madness of football. I told the lads that we are over-due a win. We need the rub of the green that sometimes changes.
"In this match it went Birmingham's way. We have only lost one in six games but we have had four draws.
"We have to challenge to turn those performances into wins. I thought the lads deserved a win because I thought it was a thorough performance.
"The style of play and the way they delivered it I thought was excellent.
"We scored two good goals and a third good goal but unfortunately the referee decided to pull up immediately. The referee told me, 'It was a really bad tackle and I jumped on it.'
"He knows he should have allowed the move to continue."