Insigne had Napoli's first concrete chance of the match just before the half-hour mark, but his effort cannoned off the post as the passage of play was cut down for offside.
The Italians were very much on the ascendency when they opened the scoring on 32 minutes, Zielinski pouncing on the rebound from an Insigne shot to fire into the net for the breakthrough.
With five minutes to go in the match, Napoli drew level on aggregate as Jose Callejon's cross was volleyed home by Insigne to set up a nervy finish for Leipzig.
Callejon then had the chance to win it in injury time, but his effort sailed over the bar as the hosts saw the final few minutes out and progressed to the last 16 of the competition.
RB LEIPZIG (4-4-2): Gulacsi; Laimer, Konate, Upamecano, Bernardo; Sabitzer, Kampl, Demme, Bruma (Forsberg 73'); Poulsen, Werner (Augustin 85')
NAPOLI (4-3-3): Reina; Maggio, Tonelli, Albiol, Rui; Allan, Diawara (Jorginho 82'), Hamsik (Callejon 66'); Zielinski, Mertens, Insigne