Bridge, who was attracting interest from Championship side Reading, successfully underwent a medical at the Stadium of Light and is available to play against Norwich on Wednesday night.
"I'm hoping that with Wayne's undoubted experience and ability he can be a big help to the team until the end of the season," boss Martin O'Neill told the club's official website.
"I'm very pleased to have him here and with the number of injuries we've sustained recently I'm sure he'll be a great help to us."
Bridge had made just one appearance for City this season in the Carling Cup after spending the second half of last term on loan at West Ham United.
The 31-year-old, who joined City from Chelsea three years ago, had not made a Premier League appearance for the club since December 2010.
Last summer's arrival of Gael Clichy further reduced his opportunities at the Etihad Stadium and manager Roberto Mancini had been keen for Bridge to find a new club before the transfer window closed.
Bridge told SAFC TV: "I'm delighted to make the move. I have met the manager and the lads and everyone seems great.
"I'm just excited to start playing football and playing a few games. The club is on the up - the manager has come in and done very well.
"Just coming in and speaking to him, everything has been positive. I have trained with the lads and they are a great bunch who are feeling really positive.
"Hopefully I can help the club progress even more."
Bridge's only game this season came in a Carling Cup third-round tie against Birmingham on September 21, and he admitted he cannot wait to get started on Wearside.
He said "I'm grateful for the chance and want to go out and work hard.
"I have looked at the fixtures and this month's quite busy with a game every few days. That's going to be good for me."
