Baum stepped in after Dirk Schuster was fired in mid-December after a run of one win in nine matches. Baum, 37, led his side to a 1-0 win over Borussia Moenchengladbach and a 1-1 draw at Borussia Dortmund.
"We are convinced that Manuel Baum is the right coach to implement and further develop the club's philosophy," said sporting director Stefan Reuter.
"Right from his first days as a head coach, his team has been meticulously prepared."
Baum, who previously coached SpVgg Unterhaching in the third tier, joined Augsburg this year to coach the reserve team.
Augsburg, one of seven clubs to have changed coach since the start of the season, are 12th in the Bundesliga with 18 points, five clear of the relegation playoff place.
They reached the Bundesliga for the first time in 2011 and since held their own.
(Writing by Brian Homewood in Berne, editing by Neil Robinson)
