Andre Hahn notched a brace as Augsburg scored four goals away from home in a Bundesliga match for the first time and extended their unbeaten league run to a club record eight matches.
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.
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.
Later in the day, Schalke 04 struck twice late in the first half to secure a 2-0 home victory over Hannover 96.
Jefferson Farfan's 39th-minute opener was followed by a Max Meyer goal give minutes later to solidify the Gelsenkirchen outfit in fourth placed after Borussia Moenchengladbach were beaten at home by Bayer Leverkusen on Friday night.
- Sports & Recreation
- Vedad Ibisevic