Osimhen put Napoli ahead in the fourth minute after latching onto a pass from Khvicha Kvaratskhelia and flicked it into the net.
The 23-year-old made it 2-0 in the 19th minute when he connected to another pass from Kvaratskhelia and fired the ball behind Sassuolo goalkeeper Andrea Consigli.
Kvaratskhelia continued the rout in the 36th minute before Osimhen completed his hat trick in the 77th minute by intercepting a poor Sassuolo pass and lobbed the ball over Consigli.
"Congratulations to the team, I am happy to contribute to this win. Of course it is good to have as many goals as this, I want to keep the momentum going," Osimhen told DAZN.
"This season the coach has given me as much confidence as I can get. I am really happy to rely on him giving me the maximum confidence. I have just returned from five weeks out, which is not easy, and it is important to get really integrated with the team."
Sassuolo ended a miserable day with Armand Lauriente being sent off in the 84th minute after he picked up a second yellow card.