Stoke took the lead on 28 minutes when Eric Maxim Choupo-Moting brought down Xherdan Shaqiri's 40-yard pass and scored.
Pascal Gross levelled on 44 minutes after Davy Propper beat four players and put in a cross.
But in first-half stoppage time Kurt Zouma restored Stoke's lead when he headed in his first goal for the club.
Brighton's Mat Ryan saved from Ramadan Sobhi and on the hour mark they equalised again.
Jose Izquierdo fired in from close range to earn a point for the Seagulls, who remain unbeaten at home since Matchweek 1.
Brighton: 25 Nov v Man Utd (A)
Stoke: 25 Nov v Crystal Palace (A)