The Toffees were beaten 4-0 at Selhurst Park in Sunday's quarter-final as Lampard became the third manager to lose each of his first four away games in charge of the club and the first since 1956.
Everton lost Andros Townsend to injury after a bright start, and once Marc Guehi had headed in the opener after 25 minutes, their resistance seemed to crumble.
Jean-Philippe Mateta made it 2-0 before half-time, with Wilfried Zaha and Will Hughes doubling Palace's lead in the final 11 minutes to complete a resounding win.
Everton was jubilant on Friday (AEDT) after Alex Iwobi's injury-time goal sealed a valuable victory over Newcastle United, but it has now lost nine of its 11 matches in 2022 and conceded 16 goals in five consecutive away losses.
Just three points separate them from Watford in the Premier League relegation zone and, despite having two games in hand, manager Lampard is worried about their predicament.
When asked how big the task facing him is, Lampard said: "Huge, huge, because some things I saw there and in our last away game are intrinsic. They're in there, they're in the culture, and they don't turn with the flick of a switch. They turn with a lot of hard work and character.
"I'm certainly up for that. I believe the players are up for that, but they have to show that in games, and they have to show a reaction in games to do things better.
"We could've been much more comfortable today, as in contesting that game to the end. That's where we should be. The way that we handled bits within the game, without Palace playing particularly well, allowed them a free pass into the semi-final, as far as I'm concerned."
Lampard did not think Palace had to play particularly well in order to progress to the final four.
"Every time we concede, our heads go down and we get worse," he said. "Did Palace have to be good to beat us today? No.
"I'm not disrespecting Palace; I've come here with Chelsea teams and seen Zaha give us problems, [Eberechi] Eze give us problems, Conor Gallagher is one of the best players in the league this year. None of that happened today, and we lost 4-0.
"The dream of getting to Wembley and playing in a semi-final and final has gone. As far as I'm concerned, the biggest prize always this year was, can we stay in the Premier League? That's the focus now."
To compound Everton's problems, it appears Townsend could be facing a lengthy spell out of action after injuring his knee.
"He got his feet caught in the turf. It looks a bad knee injury. I'm devastated for him," Lampard added.
"I don't think it will be weeks. I'll reserve judgement until we find out more. It'll be more than that."