Toffees captain Neville delivered a defiant rallying cry after yesterday's [Saturday] agonising 2-1 Wembley defeat to Liverpool, which ended the club's hopes of a first trophy for 17 years.
Everton must now lift themselves for the Barclays Premier League run-in - starting with next Sunday's game at Manchester United - with a place in Europe still up for grabs.
Should they go the other way and become consumed by yesterday's [Saturday] disappointment, they risk the same fate as Bolton, who have nosedived since losing at the same stage of the competition last season and are in real danger of relegation.
Such a hangover is unthinkable for an Everton side whose financial problems were laid bare at the beginning of the campaign, of which they spent the first two months flirting with the drop.
Neville said: "We've somehow got to bounce back from this.
"Back in October, when everyone thought the club was on its knees, we've bounced back from that disappointment.
"We're in seventh position now and the sign of a good team, a great team, is when they bounce back with victories.
"This is when we need to show our mettle and we're going to do that next week at Old Trafford."
Despite losing all three Merseyside derbies this season, Everton lie a point above Liverpool and can finish above them for the first time since 2005.
Neville said: "It won't make up for it because, at the end of the day, we want to win silverware.
"But seventh position might qualify us for Europe - you just never know.
"That would be a little bit of consolation for this season.
"We need to carry on into next season, and start the season well.
"Because we do leave ourselves a mountain to climb sometimes.
"We're proud of the way we're playing in the second half of the season.
"Sometimes, you've just got to take your punishment and that's what we've got to do."
He added: "We're all devastated, really.
"This is a special place to play football. You can tell by the reaction of the fans, it's a special day out. But it's only a special day out if you win."
It was certainly a special day out for 62 minutes, with Everton in dreamland after Nikica Jelavic's fifth goal in five games had given them a half-time lead.
But a horrendous backpass gaffe from Sylvain Distin gifted Luis Suarez an equaliser and David Moyes' men simply failed to recover before Andy Carroll's late winner.
Neville said: "I thought we had Liverpool just where we wanted them.
"We scored the early goal and that's when we really needed to score the second, really stick the knife in. We didn't do that.
"We expected them to come out in the second half and throw the kitchen sink at us.
"There was a period where we thought we'd just weathered the storm and then the first goal that they scored was a sucker punch.
"They had the momentum then and we were hanging on.
"At the end of the day, they probably deserved to win."
Goalkeeper Tim Howard said: "We were our own worst enemy.
"The disappointment comes from that there was a lot more that we could have done.
"If we could have seen ourselves through the period where Suarez scored, it would have been a different game.
"We didn't play a whole lot better after that.
"That's when you have to be a man and stick your chest out. We didn't do that and it cost us dearly.
"Sylvain is devastated, there is nothing you can say that helps someone when that happens.
"It is catastrophic and I have been there. Any pat on the back is in one ear and out the other.
"But he's a man and he will get over it, as will we all."