Boateng was sent off in the 28th minutes for pushing opponent Christian Schulz while arguing after a foul on Hannover's Sergio Pinto, with Hannover leading 1-0.
The referee had consulted the fourth official before showing a red card but Bayern officials criticised the decision as "unfair" and "inexplicable."
The German Football Association (DFB) said in a statement that Boateng had been found guilty of violent behaviour in "a less serious case" and gave him the minimum sentence.
Earlier on Monday Bayern CEO Karl-Heinz Rummenigge had demanded no suspension for the Germany international, blaming Hanover players for his dismissal.
"This was a decision by the fourth official. That is inexplicable," Rummenigge told reporters. "Perpetrator and victim got mixed up."
Rummenigge accused Pinto of acting. "It all started with his acting. He is famous for it in the Bundesliga."
Hannover ended Bayern's string of eight consecutive league clean sheets with a 2-1 win. The Bavarians, however, lead the Bundesliga standings by three points from champions Borussia Dortmund.
They are also top of their group in the Champions League.
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