Sky Sports News understands the transfer fee for the right-back could rise to £8m.
The Foxes beat off competition from other Premier League clubs, including Aston Villa, to land Justin, who was named in the Sky Bet League One team of the year last season as Luton secured promotion to the Championship.
The 21-year-old is Brendan Rodgers’ first signing as Leicester boss since taking over at the King Power stadium 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,” Justin told LCFC.com.
“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.”
Justin made 52 appearances for Luton last season as they won League One and completed back-to-back promotions in the process.
He will join up with his new Leicester team-mates on Monday as they begin preparations for the new Premier League season.
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