The Toffees overcame Stoke 1-0 on Saturday evening courtesy of a wonderful Kevin Mirallas solo effort, meaning that the team moves within four points of Chelsea in fourth.
Despite his admission that Champions League football next season is a distinct possibility, Moyes has outlined his desire to wait until the season draws to an end before seeing how to best "move forward".
"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," he told ESPN.
"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."
The long-serving Everton boss was also keen to emphasize the importance of the win over Stoke as he lauded Mirallas' contribution to proceedings.
"We've not had a great record against Stoke City recently so it was important we got the three points," he continued. "It was a great goal. Against Stoke sometimes you need individual players to win you the game because they are really well organized 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."