The German champions are understood to have paid a loan fee worth around sixteen and a half million euros, while a permanent transfer could be triggered for sixty-five million euros if Jackson reaches a set number of appearances.
Bayern's board member for sport, Max Eberl, said the 23-year-old had made his desire to join clear from the start. "Nicolas was passionate about playing for FC Bayern from the very beginning, which makes us even happier that it finally worked out," Eberl said. "Despite his young age, he has already gained a lot of experience at the top international level and shown what qualities he possesses.
"Nicolas fits our requirements perfectly with his dynamism and presence. He is hungry, will expand our offensive options with his abilities, and will immediately be an asset for our team."
Jackson, who scored 14 goals across all competitions for Chelsea last season, said he was proud to be joining one of Europe's biggest clubs.
"I'm very happy to now be part of this amazing club," he said. "FC Bayern are one of the best clubs in the world: everyone knows the names of the legends who have played here and understands that this club stands for the greatest achievements.
"I have big goals and dreams here and will give everything to win more titles with Bayern."
The move comes after Bayern's opening Bundesliga fixtures exposed their need for greater depth in attack, with injuries limiting Vincent Kompany's options. Jackson will compete with Harry Kane for a place in the forward line, adding pace and versatility to the squad.
Bayern are also preparing for their Champions League campaign later this month, where Jackson is expected to play a key role as Kompany looks to refresh his attack.