Gus Poyet's side have gone six games without a win in the league and sit bottom of the table with 10 points from 17 matches.
Mannone was pleased to keep a clean sheet against Norwich City at the Stadium of Light on Saturday, but disappointed not to take all three points as Sunderland were held to a goalless draw.
The point leaves Poyet's men three points adrift of fellow strugglers Fulham and Crystal Palace, although the Italian is remaining positive that Sunderland can stay up as the side travel to Goodison Park on Thursday looking for a first victory at Everton since 1996.
"You need to keep believing because that's what takes you out," he told the Shields Gazzette.
"We beat Man City, Newcastle and Chelsea, so I don't see why we should give up the fight.
"We'll keep putting the full effort in. We might not win against Norwich, but we might win against Everton. You never know.
"Every game is a final. It was before, but even more so now.
"You have to realise that it's our life. If I'm going down, then I'm going down and my team-mate is too.
"We need to be better than to come out of this situation because otherwise we're all going down - us, the fans, the club, everybody."
‘After Manchester City’s recent form, maybe they’re the underdogs against Manchester United!’ Former Red Devils defender on this weekend’s derby
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
