The Eagles came into the break 2-0 up, thanks to goals from Annabel Blanchard and Abbie Larkin.
Blanchard put Palace ahead after a cutback from Molly-Mae Sharpe, before Ashleigh Weerden's cross was nodded in by Larkin.
Elise Hughes then put the icing on the cake in the second-half, after she controlled Weerden's pass and finessed an effort beyond the Birmingham 'keeper.
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