Looking to make it back-to-back league wins for the first time since late January into early February, Napoli fell behind in the 22nd minute after Gerard Deulofeu arrowed a strike into the net.
The hosts looked like a different side in the second half, however, and turned the match on its head as striker Osimhen backed up his double in last week's 2-1 win at Hellas Verona with another brace.
His first was a bullet header to level the match seven minutes after the break, before a clever near-post finish 11 minutes later completed the turnaround.
Pablo Mari's late red card ended Udinese's hopes of grabbing an equaliser at the Diego Armando Maradona stadium, as Napoli saw out a crucial win in the race for the Scudetto.
The victory moved Napoli onto 63 points, level with Milan at the top of the table ahead of their trip to Cagliari later on Saturday. Udinese stay 14th.