After just three minutes of the quarter-final second leg at Stade Louis II, Dortmund goalkeeper Roman Burki could only parry a powerful long-range drive from Benjamin Mendy.
Teenager Mbappe - who scored twice in the first leg after netting in both legs of the last-16 tie with Manchester City - was in the right position to tuck home the rebound to extend Monaco's advantage to 4-2 on aggregate.
The precocious 18-year-old, who has hit 12 Ligue 1 goals this season, also became the youngest player in Champions League history to score five goals in the tournament as he continues to announce himself on the global stage.
Raul, a three-time Champions League winner with Real Madrid, was 41 days older than Mbappe when the Spaniard scored his fifth goal in the competition.
Mbappe also becomes the sixth player to score on his first four starts in the Champions League, and the first to complete the feat since Diego Costa in 2014.