The Blaugrana fell behind to a Dominik Wanner strike after just six minutes but Mboula was on target to help his side complete a fantastic comeback.
Carlos Perez, Abel Ruiz and Seungwoo Lee all found the back of the net to put Barca 3-1 ahead in the tie before Mboula stepped up with a piece of individual genius.
Collecting the ball down the right-hand side of the field, Mboula danced past one Dortmund defender before embarking on a solo run which Lionel Messi would have been proud of.
Mboula showed brilliant pace and close control to drive into the penalty area before slotting calmly into the back of the net to seal Barca's passage into the last eight.
Barca's U19s, coached by Gabri Garcia, will now take on Porto in the quarter-finals of the competition.