Kevin Mirallas' stunning solo goal in the first half settled an often dour contest and saw Everton move to four points adrift of Chelsea in fourth place.
Moyes insists they remain outsiders for UEFA Champions League football next term with Tottenham and Arsenal also in contention and would not be drawn over his future, with his contract due to expire in the summer.
He told ESPN: "I've always said we want to see how the club do and see how we can move forward together and the best time for us to discuss that is at the end of the season.
"We are the big underdogs, we need results to go for us as well as us winning our games, but we are in there and it is a great thing to say Everton are in with a chance of the Champions League.
"We deserve to be there because of how well we've played this season."
Moyes, who switched to a 3-5-2 formation with Steven Pienaar and Marouane Fellaini ruled out through suspension, was impressed with the contribution of Mirallas as he ran from deep inside his own half to beat Asmir Begovic.
He added: "We've not had a great record against Stoke City recently so it was important we got the three points.
"It was a great goal. Against Stoke sometimes you need individual players to win you the game because they are really well organised and hard to break down.
"I think he must have run 70 yards to finish it off. He has given us something else. We changed how we played today as we had lost two important players and thankfully we got the points."