Steven Pienaar scored the only goal of the game to keep the heat on fifth-placed Tottenham who dropped points at the DW Stadium, whilst the Cottagers failed to take an opportunity to enter the top half of the Premier League table.
There are just three points between Everton and London side Tottenham, with the latter having the advantage of a game in hand, and Moyes insists his players must take maximum points from the remaining games to stand a chance of overtaking Spurs.
The Scot told reporters: “All we can do is try and win our games, that’s what we’ve said. If we can win our games and see how we do; we’ve beaten last year’s total with those points today and hopefully that in itself has shown a sign of progress.
“We’ll try to keep it going to the end of the season, and if we get into Europe we’ve certainly earned it because the sides above us are all really good sides in their own right.
“People were saying we couldn’t afford to lose any more games [to get into Europe], and last time we lost it was a bit of a downer. If we’re going to have any chance we’ll try to win the next games and see where it goes.”
In regards to Pienaar’s performance, Moyes believes it was one of his best, despite still showing effects of his injury woes earlier this season.
He added: “I think since his injury he’s just been off a bit, but he played well today. I thought Steven knitted us together and obviously scored a good goal as well, so that’ll do him a world of good.
“It was a terrific footballing goal. We played some good football at times.”
Quizzed on his future, the veteran manager insists he will not decide whether he will stay at Goodison Park after the summer until the end of the season.
Asked if he’ll decide on a new contract sooner rather than later, the Toffees boos replied: “No, that’ll be at the end of the season.
“I think when you’ve been here 11 years, you’d hope you’d be given that time and respect to make that decision.”
Fulham manager Martin Jol was understandably downbeat about the result and admitted Everton simply had more of an appetite to win the game.
The Dutchman said: “From the start you saw that they (Everton) were eager, they were hungry.
“They were playing with two up front, they caused us a problem with [Steven] Pienaar on the left and they scored a good goal.
“But it was only 1-0, so second-half, we played a bit better in possession and with a bit of luck we could have had a point, but they could have scored at least two or three as well.”
Veteran goalkeeper Mark Schwarzer was the Cottagers’ stand out player, making numerous saves to keep his team in the game, but Fulham have recently liked with a number of shot-stoppers, including Roma’s Maarten Stekelenburg. However, Jol was dismissive of such rumours.
He said: “You saw today he [Schwarzer] is a fantastic goalkeeper, so we’ll try to keep him. There are a few goalkeepers linked with us; [Asmir] Begovic from Stoke City and a few other ones, but it’s not true.”