The Frenchman replaces Craig Shakespeare, who lasted less than eight months as successor to Claudio Ranieri - the coach who delivered the Premier League trophy so unexpectedly in May 2016.
"Upon meeting Claude, his attention to detail, knowledge of our squad, understanding of our potential and his vision to help us realise it were extremely impressive," Leicester vice-chairman Aiyawatt Srivaddhanaprabha said.
"He quickly emerged as the outstanding candidate and I am delighted we will have the opportunity to reinforce our long-term vision, aided significantly by his expertise."
Puel, who has been out of work since being dismissed by Southampton in June, takes charge of a team that has taken nine points from nine games and is 14th in the 20-team Premier League.
"The opportunity to help the club build on its remarkable recent achievements is a truly exciting one and I'm looking forward to working with the owners, players, staff and supporters to deliver further lasting success," Puel said.
Michael Appleton, who has won both games as caretaker manager, will remain assistant coach. Leicester's first game of the new era is on Sunday against Everton, who are searching for a new manager after firing Ronald Koeman, who was Puel's predecessor at Southampton.
Puel was sacked by the south-coast club in June despite leading the club to their first major final in 38 years when they were denied the League Cup by Manchester United.
As a tough-tackling midfielder, Puel spent 17 years at AS Monaco before starting his management career there and winning the French title in his first season as coach. Puel moved across France for a six-year spell at Lille, where he finished runner-up in 2005. Spells at Lyon and Nice followed before moving to Southampton in 2016.
"When we began the process of identifying a new manager, the board quickly established the profile of candidate we needed to take the club forward and Claude Puel was a perfect fit," Srivaddhanaprabha said.