Cotterill - who was on Redknapp's backroom staff last season before leaving during the summer - has penned a two-and-a-half-year deal at St Andrew's, while previous caretaker boss Lee Carsley has agreed to become his assistant.
CEO and director Ren Xuandong told the club's official website: "A lot of quality candidates applied for the manager position and we underwent a very, very careful and meticulous process of selection. We have every faith in Steve Cotterill to help us achieve the ambitions we all share.
"We are also delighted that Lee Carsley will be Steve's assistant manager and, with the rest of the management team, we are excited about the future prospects, we are sure they can do a great job."
Carsley will remain in charge for Saturday's game with Hull City, with Cotterill officially taking over on Monday.
Birmingham currently sit in 22nd position in the league table, although the West Midlands outfit have recorded four points from their last two fixtures.