The Black Cats slipped back to the foot of the Premier League table in midweek, with Gus Poyet's men unable to offer a positive response to Crystal Palace's victory over West Ham United on Tuesday.
They came close to doing so on Wednesday, but fell agonisingly short against an Eden Hazard-inspired Chelsea outfit.
A 4-3 reversal on home soil means Sunderland remain five points from safety, but there are still 24 games to come and Dossena insists heads are not dropping at the Stadium of Light just yet.
He told the Sunderland Echo: "We're frustrated because we scored three goals and came away with zero points. But this team has great character.
"Okay, we lost yesterday, but there is a long way until May and the other teams are not a long way in front of us.
"We've got another chance on Saturday to close the gap on the teams in front of us.
"We kept going last night because the mentality here is perfect. The fans are behind us and we want to win for them. At the end of the day, we didn't get them any points, but the mentality here is strong."
Things do not get any easier for Sunderland, as Tottenham Hotspur visit Wearside on Saturday, but Dossena claims Poyet's side fear no-one.
The Italian added: "Hazard had a fantastic game and you don't come up against players like him or (Juan) Mata every week.
"We know Chelsea are a great team. I know Tottenham are good, but I don't think they're as good as Chelsea.
"We have to show the same heart and same character because it's a very important game."