The Toffees have enjoyed an impressive start to the 2012-13 Premier League campaign, and currently sit in fourth place with 14 points from seven games.
And Moyes, who was handed the Manager of the Month award for September, believes his side can now challenge for a coveted place in Europe's premier club competition.
"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," he told reporters.
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"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."
Despite his side seemingly having broken their well-documented habit of starting the season slowly, Moyes urged caution regarding their prospects this term."The pleasing thing is that we tend not to start that well but we have won a few games," he added.
"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."
And, while Moyes is understandably delighted with his recent managerial award, he believes it is the result of a united effort
He said: "I am really pleased and with all these awards [but] 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."