The home side got off to the better start, with Xaver Schlager opening the scoring in the 35th minute. However, the goal lit a fire under Leverkusen who dominated proceedings thereafter.
Martin Terrier levelled the scores in the 40th minute before Patrik Schick put the visitors ahead four minutes later.
While RB Leipzig produced more shots, many of them were speculative from distance, so Leverkusen ended up with far higher xG at 2.72 to the home side's 1.77.
Even so, the visitors were only able to secure the points deep into second-half injury time when a Montrell Culbreath strike put them into an unassailable 3-1 lead.
Elsewhere in the Bundesliga on Saturday, there were goalless draws between Augsburg and Werder Bremen at WWK Arena, and VfB Stuttgart and Hoffenheim at the MHP Arena.
Union Berlin took advantage of a red card to Rav van den Berg to steal a late 1-0 win over FC Koln at RheinEnergieStadion, with Andras Schafer bagging the only goal of the game.
HSV and Frankfurt played out to a 1-1 draw with goals from Albert Sambi Lokonga (19') and Hugo Larsson (26') cancelling each other out.
The most exciting viewing of the day could be found at the Volkswagen Arena where Freiburg prevailed 4-3 over Wolfsburg.
Philip Treu opened the scoring for the visitors just five minutes in before Dzenan Pejcinovic levelled matters eight minutes later.
Pejcnovic struck again four minutes into the second half to put the hosts ahead with a Vincenzo Grifo penalty restoring parity in the 56th minute.
Pejcnovic completed his hat-trick with a third strike in the 69th minute, but it would prove to be in vain as a Jenson Seelt own goal in the 74th minute made it 3-3 before Derry Scherhant struck seven minutes later to hand Freiburg a dramatic win.