Trailing 2-0 heading into the 90th minute, Emil Forsberg scored twice, including a dramatic equalizer in the 96th minute to secure the club's qualification out of Group G.
Benfica, meanwhile, are officially eliminated from contention for the last 16 after their late collapse.
The Portuguese side seemed to be in complete control of the contest heading into the latter stages, as goals from Pizzi and Carlos Vinicius established what appeared to be a solid advantage.
But Leipzig, who are playing in their second Champions League campaign, roared back at the Red Bull Arena, with their Swedish midfielder grabbing all the headlines. Forsberg slotted home his first goal from the penalty spot after substitute Patrik Schick was fouled inside the area, and he then headed in the leveler after being teed up by Timo Werner.
An injury to goalkeeper Peter Gulacsi, which triggered the lengthy stoppage that resulted in nine minutes of added time being played, is the only thing that could spoil Leipzig's momentous night.
The Germans, now sitting on 10 points, will lock up the top spot in the group if they avoid defeat against Lyon on Matchday 6.