The away side took the lead in the 39th minute, as a free-kick found the head of Simon Terodde who planted the ball into the Sandhausen net for his first league goal for the club, with goalkeeper Marco Knaller helpless.
An injury to Jean Zimmer just before half time did not help Stuttgart's cause, but they were able to double their lead in the 65th minute of the match as captain Christian Gentner scored for the second consecutive match, firing home Berkay Özcan's low cross.
The home side did receive a glimmer of hope with 13 minutes of regulation time remaining, as an error from Toni Sunjic allowed Wooten to capitalise and slot past Stuttgart keeper Mitch Langerak from close range.
A late comeback was not to be however, and the title favourites held on for a hard-fought three points which will help them in their quest to return to the Bundesliga at the first attempt.
Cagey first half led to limited chances on goal
The home side made a total of two changes to their starting 11 after they recorded a narrow 2-1 victory over top-flight outfit Paderborn in the DFB Pokal on Monday, with Philipp Klingmann and Korbinian Vollmann making way for Marco Thiede and Markus Karl respectively.
Stuttgart boss Jos Luhukay opted to field the same side that ran out 3-0 winners in their DFB Pokal match against FC 08 Homburg last weekend, although he tinkered with the formation slightly as he went with a 4-2-3-1 rather than the 4-1-4-1 he deployed in the DFB Pokal.
Despite going into the game off the back of wins in the cup, both sides were on the end of defeats in their previous 2.Bundesliga matches. Sandhausen were defeated 2-0 away to Erzgebirge Aue whilst Stuttgart were on the end of a 1-0 defeat to Fortuna Dusseldorf, with those losses leaving the two sides in 16th and 7th place respectively although fans will not be too concerned by league positions so early on in the season.
The first half was a largely cagey affair, with a large amount of fouls being committed in the middle third of the pitch by both sides ultimately meaning that referee Frank Willenborg could not allow the game to flow as spectators would have hoped.
Fans had to wait until the 23rd minute of the match to see the first clear chance from either side, as Sandhausen winger Thomas Pledl latched on to a ball that was lifted over the top of the Stuttgart defence, but he was unable to direct his shot on target as Langerak watched the ball flash a few yards wide of his goal.
Die Schwaben were on top for much of the opening 45 minutes, dominating possession although they were not able to do anything meaningful with it until the 39th minute of the game. The away side won a free-kick near the corner of the 18 yard box, and a quality delivery into the area from Özcan saw striker Terodde beat his man in the air to fire a header past the helpless Knaller, putting Stuttgart one goal ahead.
The final five minutes of the half belonged to the away side, who looked to force a second goal before the break as they pressed high into the Sandhausen half, forcing errors from their back line. Tim Knipping was the first player to go into the referee's book after his poor touch was seized upon by Zimmer, with the latter forced to leave the pitch in the 44th minute after the foul left him with a nasty looking facial injury.
Stuttgart hold on for three points despite late comeback attempt
The second half began just as the first ended, with Tobias Werner putting in an awfully poor challenge on Knipping and receiving a yellow cards for his troubles, although after seeing the replays it was perhaps lucky that the Stuttgart midfielder remained on the pitch with his studs going high over the ball and into his opponent's ankle.
Sandhausen saw a golden opportunity pass them by just nine minutes into the second period, as a free-kick in almost the exact same spot as the one which led to Stuttgart's opener saw central defender Tim Kister rattle the bar with a header as Langerak left his net wide open in an attempt to claim the ball.
The away side could easily have been shaken by this near miss, but they almost doubled their lead only a minute later as Emiliano Insua's cross found Gentner free just 10 yards away from goal, but the captain couldn't direct his volley towards goal and sent the ball flying over the crossbar.
Luhukay's men finally found that elusive second goal in the 65th minute of the match, as Florian Klein linked up with Özcan down the right flank before the Turk fired a low cross towards the edge of the box, where Gentner made no mistake as he placed the ball out of Knaller's grasp and into the left hand corner of the net.
Stuttgart had to hold on for all three points in the closing stages of the match, as Sandhuasen found a glimmer of hope with the help of Stuttgart defender Sunjic who diverted the ball into the path of Wooten for a simple finish despite the attempts of the on-rushing Langerak.
The home side pressed on in the final 10 minutes of the match in the hopes of finding an equaliser, and Langerak was forced into action in the 82nd minute as a cross from the right flank almost found the head of substitute Lucas Holer, but for the fingertip save of the Australian keeper.
Sandhausen were unable to find that cutting edge in the closing minutes of the match however, and come the final whistle it was Luhukay and his Stuttgart side celebrating as they took home all three points and moved up to 3rd place in the table, leaving Sandhausen in 16th as they now remain winless after the first three games of the season.