Dries Mertens' 82nd-minute penalty and Fernando Llorente's injury-time finish settled the Group E opener at Stadio San Paulo.
The Reds came closest to scoring a minute prior to half-time through Roberto Firmino's header as goalkeeper Adrian produced a string of excellent saves throughout the match.
Jürgen Klopp's side will now look to respond in the competition when they welcome Salzburg to Anfield on October 2.
Team news
Andy Robertson overcame a minor knock to start the affair as Klopp made three changes, with Firmino, Jordan Henderson and James Milner all starting.
First half
Backed by a noisy crowd, Napoli came close to breaking the deadlock after seven minutes, but they were denied by a brilliant double save from Adrian.