Liverpool, Jan 10 Goals from Beto and Iliman Ndiaye sent Everton into the fourth round of the FA Cup on the day club's manager Sean Dyche was sacked.
News that Dyche had been relieved of his duties came a matter of hours before the game kicked off, with Leighton Baines and Seamus Coleman taking charge of proceedings for the Toffees.
Facing League One Peterborough United, Everton scored a goal in each half, Ndiaye sealing the place in the next round from the spot deep into second-half stoppage time.
After an understandably slow start, Everton started to find their feet, with Orel Mangala denied by the woodwork as his stunning volley kissed the top of the bar and went over, The FA reports.
Their League One visitors responded and Cian Hayes almost put them into a surprise lead, with Jarrad Branthwaite clearing his effort off the line.
And four minutes from halftime, Everton hit the front through Beto. He was played in beautifully by 17-year-old Harrison Armstrong. Beto then rounded keeper Nicholas Bilokapic before sliding the ball home from a tight angle.
Peterborough searched for a way back in the second half, but it was Everton who almost made it two when Armando Broja raced away 10 minutes from time. He got past Bilokapic, who had sprinted miles outside his box, but Emmanuel Fernandez made the crucial late tackle, with the Albanian forward suffering a seemingly serious injury after an awkward landing.
They did get the second goal in the end, though, with Jadel Katongo hauling down Branthwaite in the box, and Ndiaye calmly slotting home the penalty.
Dyche, 53, departs the club after a run of five winless matches that has left them in 16th place in the Premier League table, just one point above the relegation zone. Everton have scored only 15 league goals this season; only the bottom club, Southampton, have scored fewer.
"The process to appoint a new manager is under way, and an update will be provided in due course. Under-18s head coach Leighton Baines and club captain Seamus Coleman will take charge of first-team affairs on an interim basis," the club statement read.
The statement also confirms that coaches Ian Woan, Steve Stone, Mark Howard and Billy Mercer have left the club.
Everton's next Premier League match is at home to Aston Villa on January 15.
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