Having finished 14th and 15th in their first two Bundesliga seasons after being promoted for the first time in 2011, Augsburg are eighth and in the race for European places.
Stuttgart had Vedad Ibisevic sent for lashing out in an off-the-ball incident as they continued their almost traditional mid-season slide down the table.
The defeat left them 14th with 19 points, only two clear of the relegation playoff spot.
Augsburg went ahead in the 35th minute when Arkadiusz Milik tapped home Tobias Werner's cross at the back post and Hahn increased their lead with a clinical finish two minutes before halftime.
It got worse for Stuttgart when Ibisevic was dismissed for lashing out at Jan-Ingwer Callsen-Bracker and Hahn then added a third, all before the hour.
Konstantin Rausch pulled one back for Stuttgart only for Werner to add the fourth almost immediately.
(Reporting By Brian Homewood; Editing by Rex Gowar)
