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."
Stoke have now won just once in 12 league matches and are only four points above the relegation zone despite their 13th-place position.
"I thought the last two away games, Newcastle and here, we have deserved something out of the game and it is just disappointing," said boss Tony Pulis.
"You can't fault the players' commitment. I thought we played some good football at times and I don't think the players get the credit for the football we do play.
"We had chances but the disappointing thing is we haven't put them in the back of the net.
"I have been managing for a long time. I know for a fact, in our fifth year in the Premier League, our performances have been much better than the previous four years away from home but we haven't got the results. It has been that sort of season for us where we've been closer to winning games and being in games than we have before."
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