The Toffees won two and drew another of their four matches to remain in the top four heading into October.
Moyes is now keen for his side to build on their good start and maintain their elevated position for the rest of the term.
"With the quality of the Premier League it is really hard to qualify for the Champions League but I wouldn't say we can't do it," said the Scot.
"It is possible and you have to aim for that.
"If we fall short and end up in the Europa League then so be it, but in recent years we have been pretty close, finishing sixth, seventh or eighth, so we want to do that at the very least.
"I am really pleased and with all these awards it comes as a team thing. It is something the whole team strives to achieve.
"The players have played well, they are in good form. We have had a good start to the season through August and September."
Much has been made of the uncharacteristically good start by Everton but Moyes is wary of getting too carried away.
"The pleasing thing is that we tend not to start that well but we have won a few games," he said.
"I do say it carefully as I see 10 games as a start and we are only six or seven in at the moment – we have to be careful how we view it.
"People have asked me like there has been a big change, or we have done something differently but every year you try to do things right and get off to a good start.
"We have tried doing the exact same things."