Mane opened the scoring in the 26th minute with a fine takedown and finish after turning away from Manuel Neuer, before his header in the 84th minute sealed the win.
Liverpool advanced past Bayern into the quarter-final with the victory, reaching the last eight via a 3-1 aggregate scoreline following a scoreless draw in the last-16 first leg.
With his two strikes, Mane has become Liverpool's all-time leading away goal-scorer in the club's illustrious European history.
Mane entered the match one goal behind Steven Gerrard and Roberto Firmino, but finished the game with seven total away goals in Europe and the record all to himself.
The Senegal star has saved his best for when it counts in Europe, scoring nine of his 13 Champions League goals in the knockout stage of the competition.
That percentage is the second highest of any player with more than 10 Champions League goals, trailing only Ivica Olic.
In addition to Mane, defender Virgil van Dijk was Liverpool's other goal-scorer on the night, with the Dutchman also collecting an assist on Mane's opener.
By notching a goal and an assist, Van Dijk became the first Liverpool player to both score and assist a goal in an away Champions League knockout game since Craig Bellamy against Barcelona in February 2007.
2007 - Virgil van Dijk is the first Liverpool player to both score and assist a goal in an away Champions League knockout game since Craig Bellamy in the Camp Nou back in February 2007. Swing. #BAYLIV pic.twitter.com/UXO5cNS6Rs— OptaJoe (@OptaJoe) March 13, 2019
Van Dijk's goal was assisted by James Milner, who has made a habit of providing assists in the Champions League over the past two seasons.
In fact, no player has provided more Champions League assists than Milner's 10 since the beginning of last season's competition.