The African Player of the Year, who ended a six-year stint at Liverpool to move to Bavaria in the summer, also had two goals ruled out for offside in the game played at Leipzig’s Red Bull Arena.
The Senegalese star was on target on the half-hour mark for his team’s second goal in the German season curtain-raiser.
Jamal Musiala played a perfect pass to unlock the Leipzig offside trap and release Serge Gnabry whose pass across the front of goal was tapped home by Mane.
Bayern, beginning a new era after the departure of star striker Robert Lewandowski to Barcelona, had taken the lead after 14 minutes thanks to Musiala getting the better of Leipzig’s Hungarian goalkeeper Peter Gulacsi.
The impressive Musiala was at the heart of Bayern’s third goal on the stroke of half-time when the teenager’s astute pass was drilled home by French international Benjamin Pavard.
Marcel Halstenberg headed in on the hour as Leipzig cut the deficit but it only took Bayern five minutes to restore their three-goal advantage.
Thomas Mueller had a shot only half saved by Gulacsi and Gnabry was quickest to react to the loose ball to make the score 4-1.
Another new signing Ryan Gravenberch, who arrived from Ajax, came on to replace Mueller for Bayern before Mane had a goal ruled out by VAR for offside.
In an end-to-end second half, Leipzig grabbed their second goal of the evening courtesy of a Christopher Nkunku penalty after Dani Olmo had been brought down in the area.
As the clock ticked down, Bayern brought on another new signing in Dutch centre-back Matthijs de Ligt who joined from Juventus for 67 million euros ($68.2 million).
With six minutes left, Mane had a second goal ruled out, again for offside, before Spanish international Olmo pulled another goal back for Leipzig after 89 minutes after a quick breakaway.
Bayern then had to survive a few nervy moments as Leipzig pushed for a fourth but Leroy Sane wrapped up the game with his team’s fifth goal in the eighth minute of stoppage time.