The visitors began brightly and Remi Allen almost opened the scoring when she met Melissa Lawley's cross, but struck the bar from six-yards out.
Dan Carter's volley was well saved by Birmingham keeper Sophie Baggaley as Arsenal pushed forward.
With the last kick of the match, Dominique Janssen saw her free-kick strike the bar, with Baggaley beaten.
The hosts remain third in table, five points behind league leaders Manchester City - having played a game more - while Birmingham moved equal on points with Liverpool.
Arsenal Ladies were forced to settle for a draw against Birmingham City Ladies at Boreham Wood.