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 its good start and maintain its elevated position for the rest of the campaign.
"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.
"But I am really pleased and with all these awards it comes as a team thing. It is something the whole team has strived 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 and big thanks to Barclays for giving me this award."
Much has been made of Everton's uncharacteristically good start but Moyes is wary of getting too carried away relatively early in the season.
"The pleasing thing is that we tend not to start that well but we have won a few games," he told evertonfc.com
"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.
"The year we had a good start we qualified for the Champions League (in 2005) and we have been trying to get back to that.
"We have tried doing the exact same things but you need a bit of luck and you need good players.
"Importantly [this year], we had most of our recruitment done early and that gave me a chance to go into this season with a few added to the team that finished last season."
English Premier League
An Edin Dzeko double and a late Micah Richards winner helped Manchester City secure an astonishing 4-3 win over Chelsea in St Louis.