Darmstadt's Austrian full back Gyorgi Garics headed into his own net trying to clear Christian Gentner's cross in the 68th minute and Timo Werner got another in stoppage time as Darmstadt buckled under pressure in the second half.
Garics' goal prompted a fight back from Darmstadt, who had two chances to level in the final minutes, with Stuttgart keeper Przemyslaw Tyton protecting his side's slim lead with two superb saves.
Werner then sprinted clear, rounded keeper Christian Mathenia and secured the three points for the hosts with the last kick of the game.
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Stuttgart, the 2007 German champions, moved clear of the relegation zone for the first time this season, rising to 15th place on 10 points.
Hanover 96 also climbed away from trouble after coming from a goal down to beat Hamburg SV 2-1 in the northern derby.
The hosts went ahead with Michael Gregoritsch's sixth minute goal but missed several chances to kill off the game before Hanover battled back to level with a Hiroshi Kiyotake penalty on the hour.
Hanover's Senegal defender Salif Sane then headed in a 67th minute Kiyotake cross for the winner that lifted Hanover to 14th and 11 points, a point above the relegation zone.
Champions Bayern Munich dropped their first points this season when they were held to a 0-0 draw at Eintracht Frankfurt on Friday, with second-placed Borussia Dortmund, on 26 points, closing to within five of the leaders with a 3-1 victory at Werder Bremen on Saturday.
(Reporting by Karolos Grohmann; editing by Toby Davis and Ken Ferris)
