The Scottish side needed a result and a favour from Liverpool against Ajax to retain a realistic chance of finishing third, and while Jurgen Klopp's side did its part, Rangers FC was swept aside by the Serie A leader.
Napoli took the lead after just 11 minutes, Giovanni Simeone rifling home after Giovanni Di Lorenzo teed him up, and the forward was on the scoresheet again five minutes later as he tucked home Mario Rui's cross.
Tanguy Ndombele's fierce strike from outside the box clattered off the bar before the break, while the second half saw Alfredo Morelos fail to connect from close range in the visitors' best chance of the game.
Any hope of a late comeback was crushed 10 minutes from the final whistle when Leo Ostigard was left unmarked and headed home Giacomo Raspadori's corner.
With Rangers three points behind Ajax and having lost its first meeting 4-0, Van Bronckhorst's side needs a remarkable result in Amsterdam next week to clinch third place and Europa League football.