The 19-year-old has already earned five Denmark caps, but has yet to fully establish himself with the defending Bundesliga champions.
Hojbjerg has made 13 appearances in all competitions for Bayern this season, starting six times for the Bavarians.
Despite the lack of first-team opportunities, Hojbjerg has agreed to extend his contract - which was due to expire at the end of the season - until 2018.
However, he will now hope for increased playing time with an Augsburg side that is very much in contention for European football after an impressive first half of the season.
Hojbjerg told Augsburg's official website: "At Augsburg I would like to take the next step and play matches in the Bundesliga.
"I am sure that Augsburg is exactly the right club because the club has taken a great development and the team and the environment fit.
"I would like to integrate myself as soon as possible and help the team on the pitch."
Hojbjerg could make his Augsburg debut against Hoffenheim on February 1.
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