Insigne struck the opening goal from the spot on 26 minutes after Mexico's Hirving Lozano went down inside the area under a challenge from Mardochee Nzita.
The Italy international added a second when Pietro Iemmello handled at a corner, with a penalty awarded following a VAR review.
Napoli goalkeeper David Ospina, whose blunder proved costly in Saturday's 1-0 loss at Lazio, then saved an Iemmello penalty before half-time following another VAR intervention.
Elseid Hysaj was harshly penalised for handling in the box after a cross from Diego Falcinelli hit the defender's leg before bouncing up against his trailing arm.
Holders Lazio are in action later Tuesday against Cremonese, the only second-division side left in the competition, while Juventus host Udinese in the last 16 on Wednesday.