Saints are rooted to the foot of the Premier League table after winning just one of their opening 10 games this season.
Davis believes that Saints can get their season up and running with victory over the Swans this weekend.
He told the club's official website: "Everyone wants the same thing, to start winning games, and start moving up the table and we are all determined to do that.
"We've got a home game with Swansea this weekend and that is another good opportunity to get us going and get some points on the board.
"We are still confident we can get out of this position, without a doubt. It is still early in the season but we know where we are at this point in time. We are in a fight."
Davis is the most experienced Premier League player in Southampton's squad following previous spells at Fulham and Aston Villa.