Four goals from four different goalscorers secured a return to winning ways for Julian Nagelsmann's side, who were held by Hertha Berlin last Wednesday.
Jhon Cordoba put Cologne ahead, but Patrik Schick, Christopher Nkunku and Timo Werner put the visitors in control.
Anthony Modeste blasted in a lifeline only for Dani Olmo to score Leipzig's fourth two minutes later to make certain of the win.
Cologne were on a four-game winless run heading into the match after weeks of improvement under Markus Gisdol, but they were ahead just seven minutes in when Cordoba's shot was too much for Peter Gulasci after he had nudged Elvis Rexhbecaj's effort onto the post.