Late substitute Rudy Gestede was denied by a brilliant late save from Everton goalkeeper Joel Robles as Middlesbrough were frustrated in their efforts to clamber clear of the relegation dogfight at a freezing Riverside Stadium.
Gestede rose to meet an 88th minute cross from Calum Chambers and powered a header which Robles tipped over the bar to deny Boro a first win in eight Premier League games.
Captain Ben Gibson also had a second-half header cleared off the line for Aitor Karanka's men while Romelu Lukaku and Ademola Lookman both missed earlier chances for Everton.
Boro had spent part of last week on a mid-season break in Benidorm and it took them time to re-acclimatise to the plunging north-east temperatures as the visitors had the better of an open first period.