The Seagulls' victory means they are now just two wins away from their first major trophy. Arsenal, who had won the competition 14 times previously, suffered a shock exit at Borehamwood for the second consecutive season.
Liverpool avoided a day of major upsets when they scored late in extra time to overcome Charlton Athletic. The Reds' victory secured their place in the semi-finals, while Brighton will now face either Manchester United or Reading in their next match.