Steven Pienaar scored the only goal in the 16th minute but the hosts made harder work than they should have done in winning a 20th straight league match at home to Fulham.
The Toffees are three points behind fifth-placed Tottenham, who have a match in hand, and Moyes knows their destiny is not in their control.
But he has urged his team to push on until the end of the season.
"All we can do is try to win our games," said Moyes.
"If we can do that and see how we do - we have beaten last year's total with those points today [Saturday] and that hopefully shows a sign of progress.
"We will try to keep it going until the end of the season and if we do get in Europe we will have certainly earned it because the sides above us are all good sides.
"People were saying we couldn't afford to lose any games. The important thing is to keep trying to win the next game.
"We've had to be looking at the other (team's) fixtures for a long time.
"If other teams keep winning their games there is not an awful lot we can do but if we win our games that is the big thing."
The win over Fulham was more workmanlike than anything but there were passages of quality, notably for the goal which saw Kevin Mirallas and Leon Osman combine to release the excellent Seamus Coleman near the byline and he cut-back for Pienaar to slot home.
"It was harder work than it needed to be but we played well at times," said Moyes.
"It was a terrific footballing goal. On that occasion we finished it off, it was nice intricate play and a good finish.
"I thought Steven knitted us together a little bit and he scored a good goal as well so that will do him the world of good.
"It was good from our point of view but I was disappointed we didn't create more.
"At 1-0 there was always a chance they could score. I thought our football was excellent at times but we didn't really turn that good football into goals."
Fulham boss Martin Jol admitted his side, safely ensconced in mid-table, paid for their slow start which saw them hand the initiative to their hosts early on and never really retrieve it.
"At the start you saw they (Everton) were eager, hungry, and they caused us some problems but it was only 1-0," said the Dutchman.
"They were a bit more physical and that showed in the first half.
"In the second half we were better in possession and with a little bit more luck we could have had a point but they could have scored two or three goals as well."
Jol had to replace Dimitar Berbatov after just 30 minutes because of injury but he said it was too early to say whether the Bulgarian would miss the remainder of the campaign.
"Berbatov has got a tear in his calf. He normally says he is a quick healer so hopefully not," he added when asked whether he had ruled out the striker for the final three matches.