Fellhauer found the net after just eight seconds, beating the previous record of nine seconds held jointly by Kevin Volland and Karim Bellarabi.
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The goal fell after a major error from Bayern keeper Manuel Neuer.
The 40-year-old attempted to chip the ball over Fellhauer after receiving it straight from kick-off, but instead played it directly into the path of the Augsburg player.
Fellhauer then only had to run the ball into the open net to make Bundesliga history.
