The Welshman replaced Enzo Maresca, who departed for Chelsea earlier this month.
Cooper joins Leicester ahead of the 2024/25 season, penning a three-year deal until 2027.
"I am really excited and proud to be appointed Leicester City's First Team Manager. This is a fantastic club with a rich history and passionate supporters," he said.
"I'm excited to work with such a talented squad and I'm looking forward to the challenge of achieving our shared ambitions in the Premier League."
Cooper was previously in charge of Nottingham Forest, getting them promoted from the Championship via the play-offs and keeping them in the Premier League the following season.
He was eventually sacked by the midlands club after struggling to find form during the 2023/24 campaign and has been out of work since - reportedly declining approaches from clubs in the second tier in favour of a top-flight move.
The 44-year-old also found success at Swansea earlier in his managerial career as well as in England's youth set-up, where he coached the under-16 and under-17 sides.
His first game as Foxes boss will be against Tottenham on the opening day of the 2024/25 season.
