Curtis Davies put the Blues in front and Charlie Austin then had a goal controversially ruled out - referee Eddie Ilderton blew after Kieran Trippier, who crossed for Austin, was fouled by Mitch Hancox.
Burnley did get their reward, Danny Ings levelling and Ross Wallace converting a superb free-kick, only for Nikola Zigic to snatch a late equaliser.
"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," Dyche said.
"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 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."