The 1-0 loss to Augsburg was the final straw for the 53-year-old, who informed the club's board of his decision to step down after the match.
Veh, who managed Stuttgart between 2006 and 2008, only returned to the club in May, signing a two-year contract.
But after two wins from their opening 12 matches they sit bottom of the table and club president Bernd Wahler confirmed Veh's decision on Monday.
"Armin Veh informed us last night in an interview about his decision," he said. "He is convinced that a change to the coach position is necessary to get back on track.
"We respect his decision, even though we were convinced after yesterday's game from him and his work."
In a statement Veh thanked the fans for their support during a tough start to the campaign, and took full responsibility for their disappointing form.
"The team is better than the place in the table," he said. "Nine points from 12 games are just not enough.
"For that, I am responsible.
"I thank the club's management, in particular the president, for the confidence placed in me. And it is my strong desire to thank the fans. You have given us fantastic support."
Stuttgart, who have now lost three matches in a row, will begin the post-Veh era on Friday with a trip to a Freiburg side just three points above them.