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The champions started 2015 in disappointing style with a 4-1 loss at Wolfsburg and then drew 1-1 against Schalke on Tuesday.
Normal service was resumed at Stuttgart, though, with stunning strikes from Robben and David Alaba either side of half-time securing the points.
In-form winger Robben took his tally to eight goals in his last nine games when he struck with a fine first-half finish and the Netherlands international felt Bayern's clinical finishing made the difference.
"Stuttgart pulled everyone back and defended so it was important to score just before half-time," Robben told reporters.
"It put us in a good position and made the second half a bit easier. It was time we took three points, that was very important to us.
"Stuttgart were very defensive and used their wingers as full-backs. You have to go in search of the gaps. The magic moment always comes, we always make chances - but you have to take your chances.
"We scored from our first chance and that's what you need in a game like that."
