The Toffees have traditionally been slow starters to a campaign under Moyes, but have consistently turned their fortunes around after Christmas.
And Moyes is hoping Sunday's 1-0 win at West Brom - courtesy of Victor Anichebe's late goal - will be the catalyst for a strong finish to the 2011/12 season, while Neville believes Everton should be setting their sights on a European spot.
Moyes said: "We've not been that consistent this season but, if you look at our past history, we tend to get better as the season goes on.
"We tend to go on runs in the New Year. I've got to hope this could be the start of one. Everyone is talking about it and drawing my attention to it.
"Sometimes I'd rather just go along quietly and at the end of the season say: 'Hey, we've had another good second half to the season.'
"I keep getting reminded about it but there is certainly no guarantee that it happens. Winning games in the Premier League isn't easy. Everyone is fighting for the points."
Moyes, who will have on loan striker Landon Donovan available for Wednesday's clash with Bolton, added: "We haven't won enough games to date. We probably haven't played well enough.
"Recently we have played well but not scored goals. We may need to grind a few results out. Over the years at Everton, we've needed to do that at times.
"Hopefully that might rebuild some confidence in terms of scoring some goals.
"We are unbeaten in four games and it is the sort of run you need to go on regularly if you are going to be near the top. We will try and build on it."
Neville is aware of the significance of Everton's win at the Hawthorns.
He said: "It is a massive win for us, it is a great start to the year.
"Hopefully now we can kick on and challenge for a European place because that is where we want to be.
"When you play teams like West Brom and it is so tight, you've got to take your chances, particularly away from home.
"Roy Hodgson has his team really well drilled, probably the most well drilled team in the league, with two banks of four and good pace and quality upfront.
"If you don't take your chances, and that has probably been the story for the last year and a half, then it is a battle.
"Luckily we had one chance near the end of the game and we took it."
Moyes is also hoping Anichebe can make a big impact after being sidelined for four months with groin and ankle injuries.
He said: "Anichebe is a big player for us, a really important player because of what he can offer us.
"If we can get him on the pitch often enough, we know he is a really good player.
"He has not got a lot of general fitness in him even though we are working on him. Match fitness and games are what count.
"He got in 25 minutes at West Brom, which will do him a world of good, and we will keep upping it as we go along."