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, manager Carlo 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 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.
What the managers said
Fleetwood manager Joey Barton said: "It was a tough learning lesson for some of our young players. We showed too much respect in the first period but made it a cup tie in the second half. We had Everton worried at one stage.
We've seen it down the years, a crowd getting behind the team can give you momentum. Who knows? We let them off the hook by conceding soon afterwards.
"Newcastle won 7-0 and Chelsea won 6-0. At half-time and 2-0 down you're worried if the galacticos go into overdrive. Thankfully we made a cup tie out of it.
"Now it's back to the bread and butter of the league."
Everton manager Carlo Ancelotti said: "I think the performance we had in the first half was good. We were focused, concentrated, we played well and we were 2-0 up.
"After that, we started the second half with intensity but the game wasn't controlled until the end.
"I'm pleased about the performance because it was not easy to prepare, we have a lot of games and the players who do not play a lot played tonight, they played well so the momentum is good with the team.
"When we concede goals, we're always frustrated. I think it was a general mistake not an individual mistake [from Pickford].
"It is an important competition, a short competition, I think there are three games until the final which is not a lot and we have to be ready."
Fleetwood will be back in Sky Bet League One action on Saturday when they host AFC Wimbledon (kick-off 3pm). For Everton, they travel to Crystal Palace in the Premier League, also on Saturday at 3pm, and will face West Ham at home in the Carabao Cup fourth round.
Source: Sky Sports