The Potters booked their place in the last 32 of the Europa League on Thursday with a 1-1 draw against Dynamo Kiev. They have struggled to balance the demands of Europe and the Barclays Premier League, though, and on Sunday they will travel to Goodison Park hoping to avoid a fifth consecutive away defeat, all of which have come after European nights.
"It's great that they've qualified, they've got a great record, especially with the teams they had in their group. It's a really hard thing to do and they'll be delighted to have got through with a game remaining," said Moyes.
"They've got a good squad Stoke and you can see that by the amount of changes they can make to their team. But it's never easy playing Thursday and Sunday. We know why it's done, because the Champions League takes precedence, but it's a hard thing to do.
"We've got to hope that poor form they have after Europe remains and we can try to take advantage but there's no doubt we're jealous that we're not in Europe.
"I think a club like Everton, we need to be in Europe. It was missing when I came to the club, we got it back. The way the supporters embraced it was if a relative had gone away and just come back again."
The Toffees have had their own troubles this season, played out against a backdrop of disquiet from fans about the financial situation at the club.
Successive wins over Wolves and Bolton have lifted them up the table but Moyes knows there is a long way to go before the Toffees are where manager and supporters want them to be.
He said: "We could have been fourth bottom of the league last weekend if we hadn't won and results had gone against us, so let's not start thinking we're high flying at the moment. If I win another three or four on the bounce, then I might start thinking differently.
"We're scratching around trying to find the right formation, the right players to win us enough games to push us on. Sometimes you need to find your team as you go along and hopefully we're beginning to get a little bit more of a settled side."
Source: PA
Source: PA