The Bundesliga new boys, now unbeaten after 11 games, which is something a newly-promoted Bundesliga side has never done.
They also have a three-point lead on Bayern Munich, who are at Borussia Dortmund on Saturday evening in the battle of Germany's top clubs -- dubbed "Der Klassiker".
Carlo Ancelotti's Bayern need to win at Dortmund to retain top spot on goal difference.
"From a psychological point of view we really did very well in the second-half when we were behind twice," said Leipzig coach Ralph Hasenhuettl.
After exchanging goals in the first four minutes -- Leverkusen's Kevin Kampl scoring after a minute before teammate Julian Baumgartlinger's own goal drew Leipzig level -- Leverkusen's German international winger Julian Brandt put the hosts ahead again just before half-time.
Missed penalty, missed chances
The turning point came when Brandt earned a penalty on 54 minutes only for Hakan Calhanoglu's shot to be cleared by Leipzig goalkeeper Peter Gulacsi.
Swedish international Emil Forsberg equalised for Leipzig following an impressive solo run on 67 minutes and defender Willi Orban headed in the winner on 81 minutes for the Bundesliga upstarts.
Leipzig earned their place in Germany's top flight with four promotions in seven years and were already on a Bundesliga record run for a newly promoted club by going 10 games unbeaten.
The club, sponsored by Austrian energy drink giants Red Bull, have 27 points from 11 games with champions Bayern Munich on 24 points.