Everton goalkeeper Joel Robles made a superb late save to deny Middlesbrough the chance to pull away from the lower end of the table.
The goalkeeper tipped over a close-range header from substitute Rudy Gestede to keep the scores level.
The Premier League's top scorer at the start of the weekend, Romelu Lukaku, was denied adding to his 16 goals for the season by Victor Valdes, who also stopped Ademola Lookman's shot after the break.
Lookman also missed a chance after being played in by Ross Barkley, as the match ended goalless, the 10th time this season Middlesbrough have failed to score.
The shared points keep Everton in seventh, while Boro remain 15th, two points off the bottom three.
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