The Toffees were beaten 2-1 by local rivals Liverpool in Saturday's semi-final at Wembley Stadium.
Despite getting so close to next month's final, Neville believes that his side still have something to play for.
"It's going to be difficult to lift ourselves, I can't deny it," he said. "It's going to be a test of our mettle now and it will be tough.
"But back in October, everyone had written us off. Everyone thought the club was on its knees and we were finished as a team. But we bounced back.
"We've got five games left and if we can finish seventh that might give us European football. That is something to hang on to."
Everton, who are currently seventh in the Premier League table, travel to leaders Manchester United on Sunday.