A 3-1 victory over Southampton, where the Toffees recovered from Gaston Ramirez's first goal in English football with a close-range effort from Leon Osman and two strikes from Nikica Jelavic in a 14-minute period before the interval, put them within three points of leaders Chelsea.
Everton have equalled their best points haul of 13 from six Barclays Premier League matches from the 2004-05 season - when they went on to qualify for the Champions League for the first and only time.
Moyes is keen for people to not get carried away but admits he is holding out hope of securing "a dream spot" in the top four.
"I don't think Everton managers can come in here and say 'Let's avoid relegation' but I think that is always the first route for every club and it still is for Everton," said the Scot, whose side are perennially poor starters.
"The next part is then 'Can we make a European spot and if you are very good can you make the dream spots, the big ones?
"I'd like to be in there, in the mix, during the season but I don't think Manchester United and Manchester City will be worried to see Everton there at this time of the year.
"They might be worried if in April time we slip to the top of the league."
Moyes was pleased with the way his side responded after going behind early on, especially after their disappointing Capital One Cup exit at Leeds.
"We played ever so well at times in the first half but it took us 15 minutes to get going as I felt there was a bit of a hangover from the result in midweek," he added.
"But they deserve great credit for going about their job and how well they played.
"It was nearly action every minute: it was great theatre for 30 minutes before half-time and we maybe could have scored more.
"I was clapping at the number of opportunities and chances and you could see how much the crowd are enjoying it so when you drop off (after half-time) it is quite noticeable.
"It is always a dangerous score when you are leading by two because you are worried if you concede one they are back in the game and I think we played like that in the second half."
Southampton manager Nigel Adkins rued missed chances which would have turned the match in their favour.
"I think we started the game very well and for the first 20 to 25 minutes we have taken the game to Everton," he said.
"We scored a good goal from a corner and if Jay Rodriguez had gone in and scored (when clean through) to make it 2-0 it might have been a different final reflection.
"Goals change games and with 25 minutes gone the first real passage of play when Everton put a consecutive period of passes together gave them a goal.
"The crowd got behind them, they upped their tempo and the second goal came from a counter-attack which changed the complexion of the game.
"For the 10 minutes before half-time Everton were well in the ascendancy and we weren't able to deal with it.
"Second half we came out with good composure to pass the football and if Gaston Ramirez's little chip had just gone inside the post rather than outside it could have been an interesting final period."