On a day when the Germany international led the way with a brace and an assist, the start was impressive as Hoffenheim broke the deadlock in just the 10th minute when Volland carried the ball forward on the right wing before cutting inside to find the bottom corner.
Despite the early setback, Augsburg often looked the more dangerous in the first half and they came close to an equaliser when Halil Altintop rattled the crossbar.
Baumann's error may have allowed Augsburg to equalise, but the Hoffenheim supporters were praising their goalkeeper once again when he got a hand to Raul Bobadilla's powerful header.
The save proved to be a crucial moment in the match as the visitors started to threaten on the break once again and they were gifted an opportunity to regain the advantage when Eduardo Vargas went down under pressure from Hong Jeong-Ho.
Augsburg were quick to protest that the Chilean had struck the pitch with his foot while preparing to shoot, but the appeals did not unsettle Volland, who put his side ahead once again with the penalty.
Another response from Augsburg failed to arrive as the pressure lifted from Markus Gisdol's shoulders with his first league win since being appointed before the start of the season.
The result moves Hoffenheim out of the bottom three in the Bundesliga and leaves Augsburg just two points off the bottom of the table.