Cenk Tosun put Everton back in the driving seat at 2-1 but more abysmal set-piece defending would lead to two more Millwall goals
Everton were turfed out of the FA Cup by Championship side Millwall amid fury at an obvious handball goal and familiar frustrations over the team's inability to defend set-pieces.
Marco Silva's side twice led in a tie played in sodden conditions at The Den and occasionally torrential rain but squandered their advantage within a minute or so on both occasions before one more silly foul handed the Londoners the chance to win the tie from a free kick.
Silva and Everton will be livid that Jake Cooper's controversial goal that levelled the contest at 2-2 was allowed the stand — despite this being a televised match, the Football Association elected not to employ Video Assistant Referee technology — but it couldn't mask the Blues' attacking shortcomings and their galling weakness defending high balls into the box.
An unsurprisingly poor game didn't see its first shot on target until the 43rd minute when Richarlison gave Everton the lead. Dominic Calvert-Lewin had glanced an early header wide from Seamus Coleman's cross and Richarlison had missed his kick trying to convert another centre from the Irishman as Silva's men struggled to get anything going in the final third.
That changed when André Gomes's pass found Richarlison with space to shoot from outside the area and his slightly deflected shot squirmed under goalkeeper Jordan Archer with two minutes left of the regulation 45 in the first half.
Millwall were level less than two minutes later, though. Everton failed to deal with a free kick, Cooper nodded forward for Lee Gregory and the striker looped a header over Jordan Pickford and into the net.
Everton were better in parts in the second half but had only a scuffed Sigurdsson shot to show for their efforts before Calvert-Lewin was withdrawn in favour of Cenk Tosun with 25 minutes to go.
The Turk had been on the field for seven minutes before he notched his third goal of the season. Sigurdsson played him in between two defenders and Tosun tucked his shot inside the far post to make it 2-1.
A sloppy free-kick conceded by Lucas Digne set the home side up for an immediate reply, however. Everton's defence were easily beaten in the air again and though Pickford saved the first shot well, the rebound hit Cooper on the arm and bounced over the line. Referee Michael Oliver consulted his assistant to confirm the decision and the goal stood despite Everton's vociferous protests.
Ademola Lookman was taken off in favour of Theo Walcott with 11 minutes left but the Blues' attacking efforts bore little fruit and they were heading for a replay until another ill-advised transgression in a dangerous area by Digne led to a stoppage-time winner.
Once again, Millwall prevailed as the ball was swung in, there were more blue jerseys than visiting white in the six-yard box for the second ball and Murray Wallace pounced to consign Everton's Wembley hopes to the bin for another year.
Silva will point to the injustice of the second goal but the problems in all areas of the pitch that show no signs of abating were what lay behind a performance that yielded just three shots on target and precious little of the control that would be expected of a Premier League side playing against lower-division opposition.
The painful defeat effectively ends Everton's season in late January and increases the scrutiny of the manager and a number of the Blues' expensive but under-performing players.