At first openings were hard to find as Everton defended in numbers, with only Wayne Rooney forward.
Davide Zappacosta tried to find a way around their left flank, outpacing Kevin Mirallas and then jinking around Leighton Baines, but he ran out of room before he could cross the ball.
Everton were working hard and tackling hard for the new man in their technical area but they were caught out twice as Chelsea took the lead in the 26th minute.
First they fell asleep as Willian played a corner short to Charly Musonda. And then they left Rudiger unmarked at the far post to meet Musonda’s cross and send a looping header back over Jordan Pickford and into the net.
The speed of Zappacosta again threatened to open up the visitors’ backline, Ashley Williams just getting a foot to the ball before the pass could reach Batshuayi.
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Two minutes before the interval a slick move through the centre was too quick for Everton, but when Willian played the ball out to Zappacosta, the wing-back hit the side-netting.
Everton were having to resort to fouls to stop Chelsea on occasion but then began to create chances.
First Jonjoe Kenny crossed from the right and Mirallas, unmarked, cushioned a header forward to Rooney, who saw his volley well saved by Willy Caballero.
Caballero was then too casual in kicking clear as Tom Davies attempted to close him down and sent the ball straight at Aaron Lennon. Luckily for the goalkeeper, the ball cannoned off the Everton winger and straight back into his gloves.
Chelsea, though, looked uncomfortable for the first time in the match and Rudiger just got to a long cross before Lennon could pounce.
Caballero then threw himself to his left to parry Jagielka’s headed flick from Baines’s free-kick. And then the goalkeeper produced his best save, diving to his left to deny Mirallas.
Substitute Ademola Lookman dribbled round Cahill only to fire wide, then fired a more impressive-angled shot that hit the bar. But Everton then created a chance for Chelsea. Rooney passed back without looking and Batshuayi was through.
He clipped the ball past Pickford but it did not have enough pace so he ran on to it, but collided with a post as Jagielka challenged and the ball ran wide.
Willian did beat Pickford in the 90th minute, with Calvert-Lewin pulling a goal back for the Toffees at the death as Chelsea progressed.
Chelsea (3-4-3): Caballero; Rudiger, Cahill, Christensen; Zappacosta, Drinkwater (Fabregas 62), Ampadu, Kenedy; Willian, Batshuayi (Morata 85), Musonda (Pedro 70). Booked: Rudiger. Goals: Rudiger 26, Willian 90.
Everton (4-5-1): Pickford; Kenny, Williams, Jagielka, Baines; Lennon (Lookman 73), McCarthy (Calvert-Lewin 64), Baningime, Davies, Mirallas; Rooney (Niasse 81). Booked: Williams, Davies, McCarthy, Jagielka. Goal: Calvert-Lewin 90
Referee: N Swarbrick (Lancashire).