Osimhen made the difference in a tight match in Verona, heading the visitors in front in the 14th minute after darting across his man.
Verona applied plenty of pressure after the break, with Osimhen feeling the effects of one of several robust Verona challenges, before Napoli's top goalscorer recovered to double his side's lead with 19 minutes left.
A spirited Verona refused to give in and got back into the contest as captain Davide Faraoni headed home, but Federico Ceccherini's sending off for two bookable offences seven minutes from time ended their hopes of completing the turnaround.
The victory took Luciano Spalletti's title chasers on to 60 points from 29 matches played, three points behind leaders AC Milan and two ahead of Inter, who play Torino later on Sunday. Verona stay ninth.
A home loss to Milan last week dented Napoli's hopes of a first Scudetto in 32 years, but with Osimhen back fit and in form, they have every chance of pushing the Milan clubs all the way.
"I think after the defeat to Milan everyone was upset, so we went into a difficult ground against an organised team with determination and desire to win, it's a deserved win," he told DAZN.
"In order to win the Scudetto, we must try to win every game this season. Today was a step towards our objective and I believe we can do it."