With Luciano Spalletti’s side two goals up from the first leg, the Nigeria striker’s thumping header in first-half stoppage time dispelled any lingering nerves at the Stadio Diego Armando Maradona.
Osimhen doubled the Italians’ lead on the night with a close-range finish from Giovanni di Lorenzo’s cross eight minutes after the break and, having been brought down in the box, Piotr Zielinski converted the penalty himself to make it 3-0.
“We want to enjoy the moment and look ahead. It’s a great win, we re-wrote the history of this club,” Osimhen said.
“We keep dreaming, we don’t have limits. We follow our coach and look ahead.”
With a commanding 18-point lead at the top of Serie A, Napoli head into the last eight with a real chance to compete for top prize in European football.
The build-up to the match had been overshadowed as Eintracht Frankfurt fans clashed with police after arriving in Naples despite being banned from attending the second leg of the last-16 tie.
Once the game did start, it was the visitors who had the first effort of note as Rafael Borre headed off target in the 10th minute but from there Napoli took charge.
