Ronald Koeman is under increasing pressure after Everton slumped to a 1-0 home defeat against Burnley on Sunday.
The Everton boss made five changes in an attempt to breathe life into his side, who came into the game with only one win in their last five Premier League games and sitting bottom of their Europa League group following a 2-2 draw at home to 10-man Apollon Limassol on Thursday.
But despite a refreshed line-up, with Wayne Rooney left on the bench, the hosts produced another uninspiring display, Jeff Hendrick's 21st-minute strike proving to be the winner for Burnley.
Everton, tipped to push for a top-four place at the start of the season, remain 16th with only seven points from as many games, while Burnley surge up to sixth after making it five league games unbeaten.
Sean Dyche's men have collected eight of their 12 points in four away games this season, more than they managed away from Turf Moor in the entire 2016-17 campaign.
AS IT HAPPENED: Everton 0-1 Burnley
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Oumar Niasse, brought in as one of Koeman's changes after his match-winning goals against Bournemouth last week, threatened first, shooting straight at Nick Pope.
Everton squandered a better chance in their next attack, Gylfi Sigurdsson's tame finish from 12 yards saved comfortably by Pope after Dominic Calvert-Lewin and Nikola Vlasic had taken advantage of a defensive error from Ben Mee.
But it was Burnley who struck first with their opening shot on target as a slick move that spanned 24 passes resulted in Stephen Ward cutting back to Hendrick, who shimmied past Morgan Schneiderlin and fired a low shot into the bottom corner for his first goal of 2017.
Burnley went into half-time a goal up, but felt they should have had the chance to double their lead from the penalty spot before the break when Scott Arfield went down under Ashley Williams' challenge, only for referee Jon Moss to controversially book the Canada international for diving.
Everton should have been level just after the hour mark when Niasse missed the target from Leighton Baines' left-wing cross, Koeman bringing on Rooney for Schneiderlin immediately after in a tactical change.
Calvert-Lewin was off-target with another headed attempt, but despite Burnley offering no attacking threat after the break, Everton were not creating clear chances as Koeman threw on Tom Davies and Sandro Ramirez in his final substitutions.
Koeman was unhappy to see a late penalty claim turned down when Rooney headed onto the arm of Matt Lowton, while Calvert-Lewin was denied by Pope as Burnley held on for a victory that leaves the Dutchman's future in the spotlight.