Wigan Athletic emphatically eliminated the Toffees, who have fallen behind rivals Liverpool to seventh in the Premier League table, from the competition with a 3-0 quarter-final victory at Goodison Park on Saturday.
There have been suggestions that the demoralising loss may derail Everton's season, but Neville has hailed the character of David Moyes's squad and stressed that they have not given up their hopes of securing a Europa League spot.
He told Sky Sports News: "The Champions League place is in the distance, a European place, we would take that now. It's what we have been going for all season, making sure that next season there is European football being played at Everton. That's our aim and remains our aim.
"If we can finish in the top four then so be it, but at the moment our short-term aim is to win next Saturday against Man City, to recover and bounce back and show a reaction that we didn't get on Saturday against Wigan and to attack the last eight or nine games of the season.
"It's the time where we need to stand up for ourselves. We had a disappointing February, picked up our form going into March and then last week was a major setback for us. But this team, this club, has got major character and I fancy us to bounce back."
Everton have not competed in Europe since the 2008-2009 campaign.