Albion took the lead when Orient keeper Jamie Jones miscued his attempted clearance from a back pass and gifted Glenn Murray with an open goal after just 10 minutes.
The visitors had a second goal disallowed when Murray was judged to be offside when tapping the ball home.
Brighton boss Gus Poyet was then sent to the dugout by referee Steve Cook for his protestations after a penalty was not awarded when Ben Chorley appeared to bring down Liam Dickinson.
But Orient equalised eight minutes after the restart through striker Scott McGleish after he followed up an indirect free-kick in the penalty area.
An even contest after the break failed to produce many clear-cut chances, although Brighton midfielder Elliott Bennett had a fierce rising drive spectacularly tipped over by Jones to keep out what would have been a worthy winner for Albion.