Werder, who have won just two of their last 11 league games, came from a goal down against better opponents and could have even snatched a late winner.
Stuttgart, playing across the wings with the quick Filip Kostic, went ahead with Lukas Rupp scoring at the far post after yet more good early work from the talented Serbian.
With interim coach Juergen Kramny on the bench, Stuttgart were in complete control of the game until Werder struck with Ujah in perfect position to pick up a deflected shot and level with his seventh league goal in the 71st minute.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
They then rattled the crossbar with a thundering shot from Levin Oeztunali in the 85th as they came close to snatching an unexpected win.
The result left Stuttgart in 17th place on 11 points, three behind Werder in 15th.
Champions Bayern Munich suffered a shock 3-1 loss at Borussia Moenchengladbach on Saturday for their first league defeat of the season that cut their lead at the top to five points.
Borussia Dortmund, second on 35 points, needed a last-gasp goal from Shinji Kagawa to beat VfL Wolfsburg 2-1, with the Wolves slipping 10 points behind them in fifth.
(Reporting by Karolos Grohmann; editing by Clare Fallon)
