The opening goal came after just two minutes played as the visitors, who led 1-0 from the first leg, took control of the tie.
Marseille continued to dominate and had a goal disallowed on 17 minutes, Dimitri Payet firing home from close range, but the referee awards a free kick to the visitors for a foul in the build-up.
Nevertheless, the Ligue 1 outfit ended the first half with a two-goal cushion, Florian Thauvin besting Gulacsi in the 38th minute with a half-volley from close range.
The pendulum swung back towards Leipzig on 55 minutes, however, as Keita surged into the Marseille box before backheeling the ball to Augustin, who duly curled an effort past Pele.
More twists and turns were to come as Payet got on the scoresheer on the hour mark, collecting Thauvin's pass and weaving into the box before finding the top-left hand corner of the net.
Gulacsi came up for a last-minute corner but was caught out, allowing Hiroki Sakai to burst up the pitch and find the back of an empty net from 30 yards out to make sure of victory.
The final whistle confirmed a 5-3 aggregate win for Marseille, putting them in the final four along with Atletico Madrid, Red Bull Salzburg, and Arsenal.
More to follow.