Stuttgart started brightly and almost stunned their hosts early on when Wataru Endo threaded a pass towards Thomas Kastanaras, but Janis Blaswich was equal to the Greek’s effort.
Despite making a promising start, Bruno Labbadia’s side fell behind thanks to Dominik Szoboszlai. Florian Muller could only parry the Hungary international’s free-kick into his own net after the ball bounced inside the box.
Die Roten smelled blood, and came close to doubling their advantage when Andre Silva's header clipped the outside of the post from David Raum's cross.
Aside from the goal, Die Schwaben were resolute in defence with Endo the main instigator on the counter-attack, but lacked that additional bit of quality upfront to test the in-form hosts.
Nikolas Nartey was inches away from levelling the scores within a minute of the restart, as he prodded the ball just past the far post following some indecisive Leipzig defending.
But it was Szoboszlai that provided another sucker-punch for Stuttgart, doubling Leipzig's lead with a superb volley which flew into the bottom corner.
Labbadia’s side searched for a way back into the game, and Luca Pfeiffer should have found the target with his header to reduce the arrears.
The Raum and Silva combination was in action once again, but the Portugal international failed to find the target when another header failed to trouble Muller.
However, a slice of luck finally went Stuttgart’s way when Joško Gvardiol was judged to have handled the ball inside the box, after the VAR advised referee Felix Zwayer to view the pitchside monitor.
A spot-kick was awarded, and Chris Fuhrich’s scuffed attempt from the spot beat Blaswich to halve the deficit.
An anticipated Stuttgart onslaught did not materialise, as Marco Rose's side saw out the closing stages with relative ease to secure a slender victory and move to within a point of league leaders Bayern Munich.
Meanwhile, Stuttgart left Saxony with no reward for a plucky display and remain 15th, but could find themselves in the relegation zone by the close of the weekend.
Flashscore Man of the Match: Dominik Szoboszlai (RB Leipzig)
