The 2-2 result sees the north Londoners progress to the semi-finals courtesy of a 6-3 aggregate scoreline, along with Atletico Madrid, Marseille and Salzburg.
With 19 minutes remaining Lazio still led 5-3 on aggregate but Salzburg scored three times in five insane minutes through Amadou Haidara, Hee Chan-hwang and Stefan Lainer to claim a remarkable 6-5 aggregate victory.
"These are indescribable feelings of happiness", midfielder Xaver Schlager said. "I think we missed a bit of movement to be available for our team-mates, we didn't win the duels, so it was hard".
Marseille edged ahead in the tie for the first time when inspirational captain Dimitri Payet's whipped free kick was stabbed home by Florian Thauvin before halftime.
Salzburg players salute their fans after the match.
With the Germans sending everyone forward for a corner in stoppage time, Marseille made sure of the victory as Hiroki Sakai netted on the counter-attack to complete a 5-3 aggregate win.
The shock of the day came in Austria, where Salzburg defeated Lazio 4-1 to overturn a 4-2 deficit from the first leg and go through 6-5 on aggregate.
Aaron Ramsey suffered a nasty cut to his calf.
"You could see in Europe a lot of turnovers".
The Premier League side must now await their fate in Friday's semi-final draw in Nyon, as they aim to keep their challenge for the trophy, and an automatic spot in next season's Champions League group stage, alive.