Germany forward Volland, 28, has signed a four-year contract in a move reported to be worth €15million.
He is expected to be followed out of the BayArena by Kai Havertz, who is said to be closing in on a €100m transfer to Premier League side Chelsea.
"I thank the leaders of Monaco for the trust shown. I am delighted to join this club with a great history and strong ambitions," Volland said. "I followed the first results of the team [in the Ligue 1 season]. I cannot wait to join the group and get to work to bring my qualities."
Volland accumulated 77 goals and 62 assists in 247 Bundesliga games since making his top-flight debut for Hoffenheim in 2011.
He moved to Leverkusen in 2016 and racked up 50 goals and 32 assists in all competitions over four seasons.
"Kevin showed great identification with Bayer 04 in recent years. His will and his dedication to the team and the club have always been exemplary," said Leverkusen sporting director Simon Rolfes.
"But when Kevin had this opportunity to go to an attractive club abroad, we grated his transfer request on the basis on an agreement that was satisfactory for all sides."
Monaco is third in Ligue 1 having drawn with Reims and beaten Metz in its first competitive games under former Bayern Munich and Eintracht Frankfurt boss Niko Kovac.
Volland could make his debut after the international break, with Nantes next up for Monaco at the Stade Louis II on 13 September.