Second-placed AC Milan beat Palermo 3-0 to cut the gap with Inter to seven points with five games to play.
Juventus in third trail champions Inter by nine after drawing 2-2 at improving bottom side Reggina.
Inter's defeat was only their third in the league this season while it was Napoli's first Serie A win since January and their first since former Milan midfielder Roberto Donadoni succeeded Edy Reja as coach in March.
Inter, bidding for a fourth successive scudetto and first under Jose Mourinho, had the best chance of the first half when Walter Samuel's header was expertly saved by Nicolas Navarro.
Backed by the noisy San Paolo crowd, Napoli grew in confidence but wasted the final ball until Ezequiel Lavezzi fed Zalayeta and the Uruguayan forward fired home on 73 minutes.
Milan had put Inter under a modicum of pressure earlier in the day when Kaka netted two penalties either side of in-form striker Filippo Inzaghi nodding in Massimo Ambrosini's cross.
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Europa League hopefuls Palermo had Cesare Bovo sent off early in the second half for two bookings.
"Let's enjoy this win and this league position, without forgetting those who are behind us, the season is still long," Milan coach Carlo Ancelotti said, dismissing suggestions Inter could throw away the title like they almost did last season.
Buoyant Milan host Juve on May 10 and will fancy their chances of beating Claudio Ranieri's side after they put in another average showing in Reggio Calabria.
Juve midfielder Cristiano Zanetti made it 2-2 but substitute Amauri should have won it for the visitors late on.
Juve will be forced to host Lecce behind closed doors next Sunday unless they win an appeal against a ban imposed after fans racially abused Inter's Mario Balotelli in a 1-1 draw last weekend.
Despite the battling point, Reggina lie six adrift of fourth-bottom Torino, who won 1-0 at home to Siena.
The tussle to avoid the drop is otherwise tight after second-bottom Lecce beat Catania 2-1 and third-bottom Bologna overcame Genoa 2-0 to dent their Champions League ambitions.
Fiorentina occupy the fourth Champions League qualifying spot after Saturday's 4-1 humbling of rivals AS Roma.
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