Rausch struck twice in the second half of a game largely dominated by the home team, who could just not find a goal to round off their almost constant pressure.
It ended a run of 11 away games without a win for the Reds, who move back into contention for a place in Europe next season as a result, while leaving Augsburg's Bundesliga future hanging by a thread.
Augsburg started with the confidence of a team who, considering only the results of 2013, would be third in the Bundesliga standings.
Torsten Oehrl headed their first chance off target while Ron-Robert Zieler did well to keep out Jan-Ingwer Callsen-Bracker's direct free-kick in the 32nd minute.
Soon after, Dong-won Ji's shot grazed the crossbar before Callsen-Bracker was denied again with a header which Rausch cleared off the line.
Rausch then showed his value at the other end of the pitch to put Hannover ahead against the run of play on the hour mark.
Three shots were blocked as the ball pinged around the penalty area before it landed at the feet of Rausch, who made no mistake.
Augsburg did not accept defeat without a fight, though, and Zieler had to be at his best to keep them at bay before Rausch struck again in stoppage time to end the home team's hopes.
With Augsburg committing too many men forward, a long ball from Sebastien Pocognoli sent Rausch through, and he chipped the ball over Mohamed Amsif to rubberstamp victory.