It was the seventh consecutive game that Bayern conceded a goal, and their defensive weaknesses were repeatedly exposed by the quick-breaking Stuttgart team.
Philipp Forster should have doubled Stuttgart's lead when he shot wide from three metres, and Neuer was then lucky to be awarded a foul after losing possession in the box, allowing Forster put the ball briefly in the net.
Instead, Bayern's Coman punished the home side, hitting them on the break to level in the 38th.
Bundesliga top scorer Robert Lewandowski, who had hit the post with a third-minute header, then drilled in his 12th league goal of the season on the stroke of half-time from a Coman assist to turn the game around.
Stuttgart kept carving out chances in the second half and Neuer did his best to keep the visitors in front. It was not until Douglas Costa's low shot in the 87th made it 3-1 that they were able to breathe more easily.