The match was decided by Kevin Mirallas' scintillating 70-yard run and finish in the first half and built on the morale-boosting win over champions Manchester City a fortnight ago.
It lifted the Toffees to within four points of fourth-placed Chelsea and, with Tottenham in third to play next week, there is further opportunity to put a dent in their rivals' chances.
"I am nearly as pleased with that result as I was with the one against Manchester City a fortnight ago," said Moyes, whose side had not beaten the Potters since October 2010.
"We have not got a great record against Stoke so it was great to get over the line.
"It is great that we are going into April and people are still talking about Everton having a chance of a Champions League spot.
"Okay, it is a big call and a long way away but we are in with a chance and what we will try to do is finish as high up the table as we can.
"We believe we are in there. I don't think we have ever been out of it.
"We are hanging in there and are underdogs in terms of getting it, but we have those teams to play and we will go and give it a go.
"We have been close all season, we have matched most teams this year so we will try to do it again. Spurs are in a much stronger and healthier position than us, but we have been there or thereabouts most of the season and we'll not be too far away."
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