Roberto Martinez's side again struggled as goals from Victor Moses and Mame Biram Diouf sealed a routine victory for Mark Hughes' sideEverton's poor league form continued as Stoke City ran out comfortable winners at the Britannia Stadium.
After a quiet start to proceedings, Victor Moses broke the deadlock with a bullet header as he got on the end of Phil Bardsley's cross.
Everton threw on Arouna Kone in a bid to find an equaliser, but the former Wigan striker was denied by Asmir Begovic midway through the second half.
But the Potters sealed the points late on as Mame Biram Diouf tapped in after Marko Arnautovic hit the post.
Stoke were without midfielder Stephen Ireland, who suffered a gashed calf in the 1-0 win over Hull City on Saturday, while Peter Crouch's reward for coming off the bench to score the only goal in that game was a starting berth.
Everton midfielder Darron Gibson made his first Premier League start since May 2013 following his recovery from a knee injury and defender Leighton Baines failed a fitness test on a quad injury, so Luke Garbutt continued at left-back.
Garbutt set up Gibson for a long-range shot that was easy for Begovic to deal with before Gareth Barry picked up his 10th yellow card of the season, meaning a two-game ban, when he fouled Charlie Adam and was cautioned by referee Mark Clattenburg.
Stoke, already without Ryan Shawcross, lost fellow central defender Philipp Wollscheid to injury with 15 minutes gone.
With just under half an hour gone, Stoke went close to breaking the deadlock.
Moses cut in from the left and fired in a right-footed 25-yard shot that Tim Howard kept out with clenched fists.
Stoke were on top and a minute later went ahead.
Charlie Adam fed Bardsley on the right and his fine cross was headed into the top-left corner by Moses.
Steven Naismith almost worked an opening, but he was kept out by the Stoke defence and then Aaron Lennon saw his shot smothered by Begovic as the home side remained in control to the interval.
Romelu Lukaku ran clear early in the second half, but his shot was across goal and did not trouble Begovic.
Substitute Kone stung the hands of Begovic as Everton looked for an equaliser that they rarely looked like scoring, leaving Roberto Martinez to bemoan his side's continued poor Premier League form.
Diouf sealed the win when he shot home after fellow substitute Arnautovic had shot against the post.