Former Liverpool manager Rodgers won seven domestic trophies in nearly three seasons in charge of the Glasgow giants and leaves the Hoops eight points clear at the top of the Scottish Premiership.
"Brendan, 46, has signed a deal with the Foxes to June 2022 and will be at King Power Stadium for Tuesday night's Premier League visit of Brighton and Hove Albion," Leicester said in a statement.
"He will take charge of the squad for the first time ahead of Sunday's trip to Watford."
Leicester sacked Claude Puel on Sunday after a 4-1 defeat by Crystal Palace left the 2015/16 Premier League champions languishing in 12th.
"At Leicester City, the foundations are in place for us to grow and achieve many things," said Rodgers.
"Hopefully with my experiences and the talent of the players in our squad we can be successful together.
"I'm very privileged and honoured to be here as Leicester City manager and I'll give my life to make the supporters proud of their club."
Rodgers's departure with just three months of the season to go is a huge blow to Celtic, who moved quickly to reappoint a club legend in Lennon.
"I have given everything to Celtic as a player and manager already in my career and I am ready to do the same again," said Lennon.
"I will be doing all I can to continue the great work that has gone on in recent years."
The Northern Irishman won three league titles in four years during his first spell in charge between 2010 and 2014 and led Celtic to the last 16 of the Champions League after famously beating Barcelona in 2012.
"Neil has already enjoyed a hugely successful spell as manager of Celtic and, of course, served the club with distinction as player and captain over so many years," said Celtic in a statement.
"Neil will take up his role immediately and will be in charge of the team for the trip to Tynecastle tomorrow (Wednesday) against Heart of Midlothian."
Rodgers has rebuilt his reputation in Scotland after being sacked by Liverpool in 2015, just a year after nearly delivering a long-awaited Premier League title.
He joined Celtic in 2016 and has won every domestic trophy on offer since, lifting two trebles of Scottish Premiership, Scottish Cup and League Cup, as well as this season's League Cup.
However, not even the lure of a third treble or breaking the Scottish record by winning 10 league titles in a row was enough for Rodgers to reject the riches and prestige of a return to the Premier League.
"Brendan is a football manager of the highest quality and we are very disappointed to see him leave the club," said Celtic chief executive Peter Lawwell in a statement.
"This is an opportunity he wished to pursue and that is something we have to respect.
"He gave the club so much and has created so many wonderful memories for the club and our supporters. His achievements are historic."
Despite his success in Scotland, Rodgers's reign will also be remembered for a lack of progress in European football.
His first match in charge of 'The Bhoys' was a humiliating 1-0 loss to Lincoln Red Imps of Gibraltar in the first leg of a Champions League qualifier.
Celtic did make it to the group stage of the Champions League in his first two seasons in charge, but suffered club-record defeats in a 7-0 thrashing away to Barcelona and 5-0 at home to Paris Saint-Germain.
Defeat by Valencia last week saw the Hoops eliminated in the last 32 of the Europa League for a second straight season.