The Seagulls looked to be cruising after a penalty by Adam Virgo and a low shot from Joe Anyinsah put them two up after only 14 minutes.
But Ryan Jarvis headed home in the 21st minute and Adam Boyd's 71st-minute penalty deservedly pulled Orient level.
Jarvis missed the opening penalty of the shoot-out and Robbie Savage also saw Brighton's first spot-kick saved.
Orient were leading 3-2 on penalties when Virgo's kick was saved by rookie keeper Jamie Jones, but it was ordered to be retaken and Virgo scored.
Brighton squeezed through after Luke Ashworth's penalty was saved by Michel Kuipers and then Adam El-Abd stepped up to convert the decisive kick.