Free-scoring Atalanta, last season's top-scorers with 98 goals, had netted 13 goals in winning their first three games but had no answer to Napoli's attacks in an extraordinary first half.
Mexico international Hirving Lozano, out of favour with coach Gennaro Gattuso last season, set Napoli on their way with two goals in four minutes.
Napoli goalkeeper David Ospina's kick upfield was misjudged by Cristian Romero and Osimhen, signed from Lille in the summer, chested the ball down and scored with a shot on the turn to net his first goal for his new club.
Romero partially made amends by setting up Atalanta's consolation for substitute Sam Lammers in the 69th minute. He won possession, charged upfield and released Lammers who clipped his shot past Ospina.
Atalanta remained on nine points from four games, one ahead of Napoli.
Napoli have won all three of their games on the pitch, scoring 12 goals and conceding one.
However, Serie A's disciplinary committee handed them a 3-0 defeat for failing to play a match against Juventus amid confusion over the COVID-19 rules and they also had a point deducted.