The Mexico international drilled in a close-range volley after 10 minutes to give Gennaro Gattuso’s side a narrow lead at the interval, but captain Lorenzo Insigne limped off with a muscular problem.
Piotr Zielinski finished off a flowing team move within a minute of the restart, before Dries Mertens was left with time and space to fire in a shot from the edge of the area.
Lozano then slotted home his second before substitutes Eljif Elmas and Matteo Politano piled on the misery for Genoa with well-taken finishes.
Genoa's preparation was hindered by positive COVID-19 tests for goalkeeper Mattia Perin on Saturday and midfielder Lasse Schone on Sunday.
The victory sends Napoli to the top of the table with six points and a +8 goal difference from their opening two matches, while Genoa are 11th with three points. (Reporting by Alasdair Mackenzie Editing by Toby Davis)