Goalkeeper Baumann made three costly mistakes to gift Hamburg a victory which left Freiburg still waiting for their first win of the season.
Maximilian Beister and Pierre-Michel Lasogga were the first beneficiaries of his mistakes before a milder error enabled Rafael van der Vaart to seal victory for Hamburg, who pulled away from the relegation zone with a second away win of the season.
Freiburg made a confident start to the game, even if they had not tasted victory in the Bundesliga all season.
Admir Mehmedi and Vladimir Darida both looked bright as the hosts controlled the opening 30 minutes, albeit without creating enough danger in front of goal, Francis Coquelin going closest with a header from five yards out.
The game turned on a moment of madness from Baumann, who raced off his line but misjudged the bounce of a long ball out of the Hamburg defence, enabling Beister to score into an unguarded goal in the 37th minute.
Two minutes into the second half, another mistake from Baumann allowed Lasogga to make it two.
Again, Baumann came out of his area but, as he shepherded the ball back into his own area, he failed to grasp it and instead only pushed it into the path of the Hamburg forward, who scored his sixth goal in the last four games.
Baumann may have wished for a hole to open up beneath him at that moment, but there was more to come.
He spilled a shot from Beister to the feet of Van der Vaart, who completed the scoring and compounded Baumann's misery to seal Hamburg's victory.