An own goal from Sakai and Ilir Azemi's late strike earned Frank Kramer's men all three points with a fine away victory.
Stuttgart, who fell asleep in the second half after dominating the first, ended the game with 10 men after Antonio Rudiger was sent off just before Bernd Nehrig missed a penalty for the visitors.
Furth's penultimate away trip in the Bundesliga took them to Stuttgart's Mercedes-Benz Arena, but they were not given much of a chance to enjoy the occasion early on.
The home side showed straight away that they were not in a generous mood, retaining the ball for virtually the entire first five minutes.
Their possession led to a dangerous cross from Gotoku Sakai which almost became a shot as it sailed just wide of the far corner of the goal.
His fellow Japanese Shinji Okazaki then went close with a header from Alexandru Maxim's cross which also just missed the target before Maxim had a go himself, just evading the far post with his shot.
The woodwork was struck in the 38th minute by Ibrahima Traore, who only had Wolfgang Hesl to beat, before the Furth goalkeeper was called into action to make a save from Martin Harnik five minutes before the break.
The story changed completely in the second half with Furth taking the lead through an own goal from Sakai, who scored as he tried to clear the ball away from Edgar Prib.
Hesl denied Harnik an equaliser in the 64th minute before Stuttgart were reduced to 10 men when Rudiger kicked out at Felix Klaus.
It could have got worse for Stuttgart a minute later when a penalty was awarded, but Sven Ulreich kept them in the game by saving from Nehrig.
They nevertheless lacked the belief to search for an equaliser and Azemi sealed victory in the 89th minute when he was given the ball on a plate by Prib and tapped it into an open goal from close range for Furth's fourth road win of the season.