The victory keeps the Toffees' Champions League dreams alive for now, but Moyes knows their destiny is far from decided.
"I think if we can go and win at Arsenal then I think you might say we've got a chance of being involved in the Champions League," he said. "If we don't, I think you're probably going to say that we've got a chance of making the Europa League.
"So no matter what happens at Arsenal, the players have given themselves a great chance. We've been doing it for most of the season. Why would it change in the last five or six games?
"I expect the players to go and keep picking up victories and trying to win every game if we can. I said to the players the job was to get this game done. I am really worried about our pitch - it is terrible and it is affecting us.
"All you can do is keep putting the ball in the net and winning games.
"There is a real resilience and toughness to the players and a clean sheet today was important."