Sean Dyche's side claimed a point at the King Power Stadium thanks to Alex Iwobi's second-half equaliser and might have taken all three if not for a string of fine saves from Leicester goalkeeper Daniel Iversen.

In a frenetic encounter, Dominic Calvert-Lewin's penalty put Everton ahead but Leicester fought back through goals from Caglar Soyuncu and Jamie Vardy, with Iwobi then equalising for the visitors after Jordan Pickford had saved James Maddison's spot kick.

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The draw keeps Everton in 19th place, a point behind Nottingham Forest, Leeds and Leicester, and extends their winless run to seven Premier League games, but Carragher feels their much-improved display shows they may yet have what it takes to beat the drop.

"It's a big point for Everton, but also the performance," he said on Monday Night Football.

"I think they were outstanding in the game. Yes, Leicester had the chance to make it 3-1, but that performance from Everton, we saw the reaction of the fans.

"It was much improved and, as the game went on, you felt like if one team was going to get a winner, it would be Everton.

"They were by far the better team and I actually think Everton's performance will worry Leeds and Nottingham Forest.

"If Everton can continue performing like that, they will get out of trouble."

Carragher did, however, acknowledge their poor away form this season, underlining the need for positive results in their upcoming trips to Brighton and Wolves.

"The big problem for Everton is still that they've got two more away games out of the next three and they've won one away game all season," he added.

"The one home game, before you get to the last weekend of the season, is against Man City.

"You've just got to hope from Everton's perspective that they are still in with a shout at that point. But I've never felt Everton will go down at any stage really.

"I think if they had lost [at Leicester], most people would have been really fearful for them, but I just look at the form Leeds are in."

Carra: Allardyce would be decent appointment for Leeds

Leeds are only out of the drop zone on goal difference and have lost four of their last five games, conceding 17 goals in the process, with head coach Gracia's job under serious threat only three months after his appointment.

Allardyce has emerged as the favourite to take the reins at Elland Road and Carragher feels the former Bolton, Blackburn Rovers and West Ham manager, who has been out of work since leaving West Brom in 2021, would be a sensible appointment.

"It sounds crazy and we never used to think this could happen in our league, but the stakes are so high I think I could probably understand it," Carragher said of the potential that Gracia could be axed.

"There is such an ill-feeling at the club right now in terms of the connection between him and the supporters. They're not happy. They showed that frustration to him at Bournemouth."

He added: "I don't think Sam Allardyce has ever come in this late. He's normally come in with a January transfer window to bring his own players in and have time to change it.

"But I think it would be a decent appointment. Whether the crowd get behind it, I'm not so sure. I don't know how Leeds supporters would react to Sam Allardyce.

"But in terms of trying to make them more solid and better defensively, there's probably no one better out there."

With daunting fixtures remaining, including home games against Newcastle and Tottenham, Carragher believes a late change of manager could be their only chance of getting the results they need to stay up.

"They've got to try to create some sort of feel-good factor in the home games they've got, to bring some kind of energy and a performance to get them over the line, so I would not be surprised at all if they made that change," he said.

"It's very difficult to instil that [a clear style of play]. The only thing I believe the board and hierarchy will be thinking is, 'Can we make Elland Road a fortress in these last couple of games at home?'

"I think that's the only thing they can do. Bring someone in where it becomes a tough game for the teams going there, where there's an atmosphere, there's an energy, because right now that has vanished completely."

Dyche salutes Everton's improvement

Everton boss Sean Dyche, meanwhile, praised his side for coming back to take a point against Leicester and hailed their improvement having been beaten 4-1 by Newcastle last time out.

"It's a big improvement," the Everton manager told Sky Sports. "There was more consistency to the performance, obviously a few mistakes - we know we have to eradicate those.

"But 23 efforts away from home and I think we had nearly 50 entries into their box. It shows the way the team are trying to progress and a much better general and consistent performance, which I asked for.

"Against Newcastle we were at least decent for 70 minutes and went under too quickly. [On Monday] we didn't. We held our nerve, kept trying to create things and kept trying to create chances."

Dyche also welcomed a goal for his No 9, saying Calvert-Lewin could be a major asset for his side in the weeks ahead.

"Dominic Calvert-Lewin looks sharper, he looks fitter and stronger again," he added. "So he will be a weapon for us over the next four games.

"He looked a threat and his true fitness, which I spoke about when I got here, is coming back.

"He looks stronger and a lot more alert."

Everton's remaining fixtures

May 8: Brighton (A) - Premier League, kick-off 5.30pm

May 13: Man City (H) - Premier League, kick-off 3pm

May 20: Wolves (A) - Premier League, kick-off 3pm

May 28: Bournemouth (H) - Premier League, kick-off 4.30pm