Les Blues forward, Kylian Mbappe had put the hosts ahead, hitting into the bottom corner on the counter-attack to register the Parisians' first goal of the night.
Lionel Messi's initial effort was tipped onto the goal post before he tapped the rebound into an empty net to level score. The six time world best player then scored a Panenka penalty to close the game.
The drama did not end without PSG winning another penalty kick deep into the injury time. However, Mbappe, rather than Messi stepped forward and hit high up over the bar to deny the Argentine a chance of registering a hat-trick in the night.
With the win, the French Millionaires jump above Manchester City to top the Champions League Group A table on seven points.
All three goals of the Argentine for PSG have come in the Champions League since he joined the club as a free agent in the summer. He also scored in their 2-0 win over Manchester City in the second game.
His two goals on Tuesday have now extended his goal scoring records to 123 Champions League goals. It also means RB Leipzig is the 37th team he has scored against in the competition.
Messi is also the fourth player to score in their first two home Champions League matches for PSG after Brazil's duo of Neymar and Alex, and current Liberia President, George Weah.
PSG only managed four shots on targets in the game with Messi recording three of that in an underwhelming performance by the club.
Despite the good show by the Germans who were unfortunate to lose, the club are yet to record any point in their three Champions League games played so far this season having lost all.
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