Leipzig, who lost their Bundesliga opener 3-2 at Bayer Leverkusen, cancelled out Stuttgart's first-half lead in the 51st minute through Benjamin Henrichs before Dani Olmo, Lois Openda, Kevin Kampl and Xavi Simons completed the comeback.
Stuttgart stunned the hosts when Leipzig's David Raum mistakenly passed the ball to Atakan Karazor who set up Serhou Guirassy for his third goal in two league games.
The Guinea international had also scored twice in their season-opening 5-0 home win over VfL Bochum last week.
Leipzig, 3-0 winners over Bayern Munich in the German Super Cup two weeks ago, levelled six minutes after the break when under-pressure Stuttgart goalkeeper Alexander Nuebel attempted to clear the ball which bounced off Henrichs and into the goal.
Olmo's superb turn and finish between two markers in the 63rd put the hosts in the driving seat with the in-form Spaniard having scored a hat-trick against Bayern in the Super Cup before adding another goal in last week's loss.
Openda's effort three minutes later gave them a two-goal cushion but Leipzig scored twice more with Kampl's outstanding shot from the edge of the box in the 74th and Xavi just two minutes later as Stuttgart capitulated.
Champions Bayern will look to make it two wins from two league matches when they host Augsburg on Sunday.
