The Gunners controlled much of the possession and territory but were unable to find a breakthrough during a competitive first half. Bukayo Saka and Noni Madueke both threatened for the visitors, while Napoli also created opportunities through their attacking players.
Arsenal continued to apply pressure after the interval and eventually broke the deadlock in the 75th minute.
Ødegaard collected the ball outside the penalty area before unleashing a powerful effort that struck the post and found the net, giving Arsenal a 1–0 lead. Christos Tzolis provided the assist after making an immediate impact from the bench.
Napoli pushed for an equaliser in the closing stages, but Arsenal's defensive organisation kept the hosts at bay. David Raya and the Arsenal back line stood firm as Mikel Arteta's side secured an important away victory.
Kai Havertz appeared to have doubled Arsenal's advantage late in the match, but the goal was ruled out for offside.
• Arsenal: Martin Ødegaard 75'
The victory gives Arsenal an ideal start to their European campaign and continues Ødegaard's impressive beginning to the season. The Norwegian captain has now scored five times or provided an assist in each of Arsenal's last five matches, according to reports following the Napoli victory.
The result is also significant for Arteta's side following Sunday's 2–1 Premier League victory over Chelsea. Arsenal have now opened the campaign with three Premier League wins and followed that run with another important victory in Europe.
For Napoli, the defeat adds to a difficult start under Massimiliano Allegri, who saw his side struggle to convert their chances despite periods of pressure.
Arsenal will return to Premier League action on Saturday when they travel to face Sunderland, before continuing their Champions League campaign against Lille in October.