Justin enjoyed a terrific breakthrough campaign at Luton last term, making 52 appearances across all competitions and scoring three goals.
A versatile full-back capable of playing on either flank, Justin's arrival marks the first permanent signing by Rodgers, who replaced Claude Puel in February.
"I'm over the moon. I feel like I'm ready for this moment. I hope to bring a lot of energy and commitment to the badge," the 21-year-old, who has signed a five-year contract, said.
"The manager is great with the younger players and developing them through their careers. Everyone speaks very highly of him and I hope he can help me too."
One player who may well be on his way out of Leicester in the off-season is Harry Maguire, with the England defender reportedly at the centre of a transfer tug-of-war between Manchester City and Manchester United.
Leicester would be loathe to lose the England centre-back, however, while Rodgers is reportedly keen to ensure Youri Tielemans, who impressed on loan from Monaco last season, returns to the Foxes.