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Max Kruse had another chance for Monchengladbach shortly after, picking up a rebound from another Ulreich stop, but he sliced his effort wide.
The away side continued to look stronger in front of goal, and were able to break the deadlock shortly before half time, when Raffael rounded Ulreich to slot into an empty goal.
Kruse set up a chance for Juan Arango early in the second half, but he fired his shot straight to Ulreich, while Raffael headed wide moments later.
However, a long ball forward for Oscar Wendt gave the defender plenty of time to control the ball before smashing it in to double Monchengladbach's lead.
Timo Werner created several chances for Stuttgart, but their opponents defended hard to prevent them from getting back into the tie late on.
The result leaves Monchengladbach seven points off leaders Bayern Munich in fourth.