Substitute Charly Alcaraz hit a late winner to give Everton a 1-0 home win over much-changed Wolves in the third round of the Carabao Cup.
The victory now tees up a meeting with Newcastle in an all-Premier League tie in the week commencing October 26.
Alcaraz's 80th-minute winner came just three minutes after coming off the bench, and it maintained Everton's unbeaten start to the season.
It was just Alcaraz's second goal for the club, having scored 13 months ago in this competition.
Everton were deserved winners over the Championship side, who made nine changes from last weekend's league win at Sheffield United.
Wolves' goalscoring hero at Bramall Lane, Raul Jimenez, was given a start, but he was concerningly taken off midway through the first half with an injury.
Wolves boss Cesar Peixoto revealed at his post-match press conference that Jimenez will be ruled out of Sunday's Black Country derby against West Brom at Molineux, live on Your Site.
Wolves debutant Joao Virginia was in inspired form against his old club as he pulled off a magnificent close-range save to deny Thierno Barry's header in first-half stoppage time.
Virginia's opposite number Jordan Pickford made a good save to deny Jimenez's replacement Adam Armstrong early in the second half, which was the visitors' first shot on target of the game.
Virginia then denied Everton substitute Brennan Johnson, but he could do nothing to stop Alcaraz's powerful curling strike from inside the area late on.
It was a goal that gave on-watching UK Prime Minister Andy Burnham and his Canadian counterpart and fellow Everton fan Mark Carney plenty of joy as David Moyes' side remain in the hunt to earn their first silverware since 1995.
Everton assistant manager Alan Irvine believes Tyler Dibling is starting to look like he belongs at the club after a difficult first season.
The 20-year-old was named player of the match in the win after comfortably his best performance of a stuttering first 13 months of his Toffees career as he provided the assist for Alcaraz's winner.
"Since he came back for pre-season, he looks like he belongs here; he looks more comfortable," said Irvine.
"He is a very quiet lad, and it can sometimes take young lads time to settle in, especially at a club like this with the expectation so that was probably difficult for him last year.
"I said to him before the game, 'You've been waiting patiently - or impatiently - and you've now got your chance, and now you must take that opportunity. Make sure the manager has a tough decision to make for the next game,' and he's done that.
"You saw a little moment of that acceleration he has got, and he doesn't show enough of those moments, and he needs to be doing more.
"I've spent a bit of time with Tyler and Tyrique (George) this pre-season trying to get them to contribute more than previously, and there are certainly signs that is coming."
Meanwhile, Peixoto did not regret making nine changes.
He said: "I think we did a good game; we competed against a strong Premier League team.
"I think today a lot of young players did a great job.
"I'm happy with the job today but not with the result."
