Leeds seemed destined for their second home defeat in four days until Max Gradel's cross was headed down by Patrick Kisnorbo for Snodgrass to score with a low left-foot drive from 12 yards.
Glenn Murray's 77th-minute penalty had given Brighton the lead against a Leeds side low on confidence after just one win from seven league games.
Murray was barged over by Leeds captain Richard Naylor just inside the box and the Seagulls striker got to his feet to beat Casper Ankergren from the spot.
Brighton had defender Adam Virgo sent off nine minutes from the end for a high challenge which sent Gradel hurtling into the air and sparked an angry reaction from Kisnorbo who was booked.
Leeds manager Simon Grayson made four changes from the team beaten by Walsall in midweek but there was little improvement until Snodgrass popped up to rescue a point for the hosts.