The 22 year-old joins Bayer for a fee of £35m and has penned a five-year contract to 2030.
Bayer director of sport Simon Rolfes described him as "one of the most promising English central defenders."
He added: "Quansah is fast, agile and a good footballer. He has already made an impressive mark in Liverpool's defence, which is characterised by absolute world-class players.
"With Jarell, our defence will gain distinctive dynamism, speed and toughness in duels and another very important building block for the Werkself of the future."
Ambitions match mine
Quansah also said: "I was able to get a good impression of the team and the club when we met Leverkusen with Liverpool in the Champions League last season.
"A great team - they played the first 60 minutes like hardly any other team at Anfield.
"In the discussions, they finally explained to me exactly what ambitious goals they have here. It's a great task and challenge for me to become part of Leverkusen's current awakening."