Neville's men lost ground in their bid for a top-four spot in the Barclays Premier League as they were held by Swansea in a goalless clash at Goodison Park.
The Merseysiders remain in a healthy fifth place but the result meant they missed the chance to gain ground on Tottenham while Chelsea pulled further away.
Yet with the Toffees traditionally strong in the second half of the season, Neville expects them to keep pushing.
The veteran skipper said: "We need to keep believing and thinking like that, it is the only way to be.
"We want to finish in the top four, and not just fourth. We want to catch Chelsea and Man City.
"We have got to keep aiming high, not waiting for teams to catch us up. We can't look behind us.
"We have got to be positive, look forward to the next games.
"Every game in the Premier League in the next month or two is going to be really difficult.
"It is that time of year, the pitches are getting softer, a bit bobblier and the weather is cold.
"It is about digging out results rather than playing fantastic football. This was a disappointing draw but we live to fight another day."
Everton had numerous opportunities to break the deadlock but did not overwork Swansea goalkeeper Michel Vorm.
Neville had one of the better chances early in the second half when he took over an attack from Seamus Coleman and forced Vorm into a low save.
Swansea earned a point through their defensive resilience and actually went the closest to scoring when Michu's attempted chip was palmed onto the crossbar by Tim Howard.
Neville, 35, said: "It was slightly disappointing because we had two or three chances.
"Against teams like Swansea, who drop a bit deeper, you need to take the individual chances.
"We didn't take them but maybe a draw was a fair result in the end.
"We need to play with more speed but we are still in a great position in the league to attack the second half of the season and we are still in the FA Cup."
Swansea were not allowed to play with their usual fluency but they maintained their momentum after impressive recent cup results against Arsenal and Chelsea.
The Welsh side held the Gunners in the FA Cup third round - and will replay this week - before beating Chelsea in the first leg of their Capital One Cup semi-final.
Manager Michael Laudrup made four changes in a bid to freshen up his side after their recent exertions and, after extending their unbeaten run to seven games in all competitions, felt it paid off.
He said: "When I am making a programme I have to think about a game and then the next games.
"If somebody has played three games of 90 minutes, for the fourth one I have to rest him.
"They are all important games - on Wednesday we play Arsenal and everybody wants to play at the Emirates.
"Then we play Stoke back in the league, very important. Then we have the second leg against Chelsea.
"After that, if we beat Arsenal, we will have another FA Cup game and then we will have two away games at Sunderland and West Ham in the league in only four days. They are all important.
"It is not to say, 'For this game I leave seven out', it just shows in this moment the confidence and spirit in the team is very good.
"It shows we can change three or four and you don't notice.
"Obviously there are different individual qualities for each player but you don't think the level has gone down. For a manager that is very important."
Laudrup is considering whether to bolster his squad during the current transfer window but is not willing to comment on speculation.
Osasuna forward Roland Lamah has been linked with a move to the Liberty Stadium.
Laudrup said: "I can say nothing is done with any player, in or out. When things are done then we can talk about the players."