Albion recovered from Steven Caldwell's early goal to lead 2-1 at half-time, but Kuszczak fumbled a Nikola Zigic header over his own line in injury-time to give Birmingham a point.
Poyet said: "We dropped two points against Birmingham and we only have ourselves to blame.
"It was really sad to see Tomasz Kuszczak make one mistake. It was terrible. Let us hope it is a one-off.
"He is devastated. I know Tomasz from last year and he has not made one mistake. But I am now going to learn from his reaction in the next game."
Ashley Barnes was one of Brighton's goalscorers, and Poyet said: "We scored two very good goals and the bonus for us was Ashley Barnes. He works well everywhere."