In a game of half-chances, Manuel Lanzini's goalbound second-half effort was the closest any of the two sides came to finding a goal.
With the share of the spoils, the point takes Everton above Arsenal into sixth-place but the Gunners have three games in-hand over their Merseyside rivals. Meanwhile, West Ham edged closer to safety as they remain in 13th place, seven off the drop zone.
Elsewhere, Swansea and Hull record survival-boosting victories, while Bournemouth romped to victory against 10-man Middlesbrough.
EPL Table | Premier League standings after Gameweek 34
Crystal Palace head to Anfield to face Liverpool on Sunday looking to cause another shock after recent wins over Chelsea and Arsenal.
Palace will have to do without loanee Mamadou Sakho, who is ineligible to face his parent club. Any positive result will all but secure Palace's Premier League status for another year.
Liverpool, though, will know a win will close the gap on second-placed Tottenham to two points having played two more games.
Manchester United will be desperate for a win to keep their top-four hopes alive, playing their game in hand on fourth-placed Manchester City when they travel to Burnley on Sunday.