The Seagulls climbed a place to 15th after the low-key encounter – seven points clear of 18th-placed Fulham who have played a game more.
Brighton pushed for a winner late on but Neal Maupay lashed an effort over from 10 yards and Lewis Dunk had a header tipped over the bar.
The point did little for Everton’s hopes of Champions League qualification with Carlo Ancelotti’s side seven points adrift of fourth-placed West Ham with a game in hand.
They were without striker Dominic Calvert-Lewin because of injury and did not force Brighton goalkeeper Robert Sanchez into a save until the 71st minute.
A win would have all-but secured another Premier League season for Brighton but a point was a decent enough result.
They offered almost all of the game’s attacking intent but, once again, were left rueing their lack of a clinical presence up front.
Of their 23 shots, Maupay had their two best chances. Had he shown more composure with a first-half effort that was deflected wide or later in the second half the three points would have gone to Graham Potter’s side.
Still, they have taken seven points from their past four games and have moved clear of trouble. One more win and they will surely be safe.