The 35-year-old was released by the Denver Broncos earlier in the off-season after 15 years with the club, but wasn't on the free-agent market for long until the Saints secured him to a two-year deal.
"We're all on the same page and we all have the same goal in mind and that's to win a championship," Bailey told reporters. "[The Saints] are very close [to a championship]. You look at, Coach Payton took a year off, came back and they got in the playoffs, and won a game.
"They're very close. I think any time you have a guy like Drew Brees calling the shots on offense, you're always going to have a shot. Then, you put a good defense in with that and you give yourself a better shot.
"They've got all the pieces, so hopefully I'm another piece to help them get to where we want to go."
The Saints reached the divisional round of the playoffs last season, where they were beaten by eventual Super Bowl winners the Seattle Seahawks.