Beausejour gave the visitors the lead on 17 minutes before Johnny Heitinga levelled terms for the Toffees 18 minutes later.
Everton made the brighter start with Leon Osman forcing a good save from Ben Foster after ghosting into the area. That was before Jermaine Beckford saw his low drive palmed away by the Birmingham keeper.
The goal came completely against the run of play. Beausejour, in space, nodded home from Jordan Mutch's cross from the right.
Everton didn't let their heads go down. Seamus Coleman saw a penalty turned down on 24 minutes. And 10 minutes before the break the home side levelled terms when Heitinga unleashed a rocket into the back of the net from Leon Osman's corner.
Everton went close to a second before half time, Foster making a top-class save to keep out Seamus Coleman from Leighton Baines' centre.
The Toffees looked the more threatening after the restart but came up against committed Birmingham defending.
Saha saw another Everton penalty appeal turned down, while the same player saw an effort blocked by Martin Jiranek.
Beckford, playing on the shoulder, kept the Birmingham defence busy and looked to be through but was put off his shot by the attentions of Roger Johnson.
The result sees Everton move up to ninth and Birmingham out of the drop zone.