The visitors were coasting at half-time through two Richarlison goals only for England’s number one to blunder three minutes after the restart.
Pickford, one of five players retained from Saturday’s victory over West Brom, dithered over a clearance virtually under his own crossbar and it was charged down to allow substitute Mark Duffy a free shot.
Alex Iwobi looked to have rescued his team-mate with a goal less than two minutes later only for more indecision from Pickford, who sat out the previous round last week to allow Joao Virginia to make his debut, to lead to another goal.
Having started to come for Glenn Whelan’s far-post cross he changed his mind and, when Ched Evans returned the ball, Pickford got a hand to Callum Camps’ overhead kick which was straight at him but could not keep it out.
For the second time in 10 minutes Ancelotti was left speechless and shaking his head in his technical area.
It made for an uncomfortable final half-hour even after Bernard fired home a fourth and substitute Moise Kean scored with Everton’s last kick of the night.
It was an evening manager Carlo Ancelotti would not have envisaged having decided to field a strong line-up with Richarlison, Pickford, Michael Keane, Lucas Digne and Saturday’s hat-trick hero Dominic Calvert-Lewin all included to get the job done.
Richarlison played his part as did Niels Nkounkou, the £240,000 summer signing from Marseille, who put in another impressive performance at left-back to match his one against Salford, with Bernard capping another fine display in central midfield with a goal.
Fleetwood boss Joey Barton, a boyhood Evertonian, must have feared the worst after Richarlison put them 2-0 up after 34 minutes, diving in at the far post to head home Bernard’s inviting cross and then blasting home from Iwobi’s clever backheel in a crowded penalty area.
But Everton’s position of comfort – they enjoyed 80 per cent possession in the first half – lasted all of two minutes and 40 seconds into the second half when Pickford’s aberration handed Fleetwood, who had not managed a shot in the first half, a lifeline.
They responded well to Iwobi adding Everton’s third with Camps scoring the goal of the night, after another Pickford misjudgement, to keep the game in the balance until Bernard and Kean settled things to set up a home tie against West Ham.
