After initially permitting Jackson to fly to Munich and take a medical with Bayern, Chelsea moved to blocked the deal on Saturday night after Liam Delap suffered a hamstring injury in victory over Fulham.
Chelsea ordered Jackson back to London, but the Senegal striker refused to budge. Talks then resumed between Jackson's agent and Chelsea in an attempt to convince the club to reverse their decision.
BILD says while the haggling was continuing, Bayern had Jackson take his medical on Monday morning in anticipation that Chelsea would agree to the original deal.
Eventually, Chelsea relented, and with his medical already passed, Bayern were able to finalise the deal in time before the deadline fell.
"I'm really happy to now be part of this great club," said Jackson. "Bayern is one of the best clubs in the world.
"Everyone knows the names of the legends who've played here and knows that this club stands for the greatest success.
"I've got big goals and dreams here and will give my all to help Bayern win further titles."