PHOTO COURTESY: SAFC.COM
Kirchhoff agreed an 18-month contract for a fee reported to be in the region of £750,000 ($1.09 million, 1 million euros).
The 25-year-old centre-back moved to Bayern from Mainz in 2013 and went on to spend two spells on loan with Bundesliga rivals Schalke.
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"He has good pedigree - he started playing first team football at 17 or 18 years old at Mainz and that he progressed to a club as big as Bayern Munich shows he has talent," Allardyce told Sunderland's website.
"He's frustrated at not being able to play a bigger part in Bayern's team in the time he's been there and he wants to experience the Premier League."
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Allardyce moved for Kirchhoff after missing out on QPR's Steven Caulker and Lorient's Lamine Gassama.
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