The 22-year-old has signed a long-term deal with the Bundesliga club, marking the fourth signing of the summer for new manager Ole Werner.
Bakayoko had been previously linked with Premier League sides Everton and Nottingham Forest but turned down their offers in pursuit of a move to Germany.
RB Leipzig will pay a transfer fee of €18 million plus an additional €4m in bonuses for the winger who scored 12 goals and provided three assists last season.
"I'm really looking forward to the new challenge at RB Leipzig. The club has made a name for itself internationally and stands for the kind of ambitious, attacking football that I love" Bakyoko told the Leipzig website.
"I've been really impressed by the way RB Leipzig endeavoured to win me over in recent years and gave me the feeling that they are fully committed to me. That, and the current discussions with all those responsible, were the deciding factors for me to move to Leipzig and the Bundesliga."