After picking up their fourth straight win against Roberto Mancini's side, the Toffees have opened a gap between themselves and the relegation zone as they look to push on from an impressive victory.
A goal each from Tim Cahill and Everton’s player of the season so far – Leighton Baines – saw the Toffees gain three points on those in the European places and climb to just six points behind fifth placed Tottenham.
Such a reasonable gap has fuelled talk that Everton could perhaps push for a European place over the Christmas period, but Moyes isn’t so sure.
"I still feel we are big outsiders when people talk about Europe," he said.
"This result keeps the Premier League exciting. I am sure many teams were happy with it, not just Everton.
"It showed the quality we have got at the club. But we have a great deal of catching up to do.
"We have been doing a great job trying to make up ground and we will keep going. But we are a bit further back than all the other teams."
Moyes admitted that his team have to start getting results against teams of a similar stature to themselves, as well as the ‘lesser lights’ of the league after good performances against the league’s bigger clubs.
“Confidence is high because we have great belief at the moment,” he said.
“We have been playing well recently. We have players who can make a difference and help us play well.
"We have already had good results this season against Manchester United and Chelsea. Now coming here to Manchester City as well means we have a bit of momentum going into the next few games."