Crofts was sent off for two bookable offences 20 minutes from time, but the Seagulls came out of the game with their first draw since Poyet took charge.
Orient winger JJ Melligan forced a good save from keeper Peter Brezovan in only the second minute, and Melligan nearly snatched maximum points for his side when his inswinging cross rebounded off the post in the 69th minute.
The visitors suffered an injury blow shortly after the break when veteran striker Scott McGleish was forced off with a leg injury following a collision with Brezovan.
McGleish should have put Orient ahead in the 29th minute following a mistake by James Tunnicliffe, but wastefully shot wide in a one-on-one situation with Brezovan.
Brighton went close just before the break when striker Nicky Forster put a glancing header over following good work between Elliott Bennett and Jim McNulty.