Midfielder Manzambi arrives after an impressive World Cup campaign in which he scored three times and provided two assists in four games before suffering an injury.
Switzerland reached the quarter-finals of the World Cup before losing to eventual finalists Argentina.
Manzambi, 20, also played a key role as Freiburg reached the Europa League final, where they were beaten by Villa.
Manzambi, who was also a target for Newcastle, was named the Europa League's young player of the season and is Villa's first signing of the summer transfer window.
"I am very, very happy and I am excited to play with this club," Manzambi told club media.
"It's a very big club in England, and they're playing in the Champions League. I've already played against Aston Villa, and for me, it was the perfect plan.
"I hope I can play the same as in the World Cup and with Freiburg."
Villa are also reportedly close to completing the signing of Wolves midfielder Joao Gomes.
Unai Emery needs midfield reinforcements after selling Youri Tielemans to Manchester United for £35 million.