Magnificent Albion lead by two goals to nil at Goodison Park.
The opener came after just three minutes from Alexis Mac Allister after good work from Neal Maupay.
The Seagulls doubled their lead on 20 minutes as Dan Burn headed in a corner at the back post.
Dominic Calvert-Lewin missed a penalty for Everton after Enock Mwepu upended Anthony Gordon.
The England striker blasted the ball high and wide.
Albion have gone in search of a third goal, with Adam Lallana and Maupay having half chances.
Brighton’s last league point at Goodison was in October 1982.
If the Seagulls win, they will go back up to eighth place.
