The hosts, level on points with third-placed AS Roma, extended their unbeaten league run to 15 matches but could easily have secured victory having wasted other good chances.
Genoa defender Dario Dainelli, a recent arrival from Fiorentina, deflected a cross onto his own post early on and Napoli centre back Paolo Cannavaro soon unleashed a powerful shot which cannoned off the bar.
Driving rain in Naples and the continued absence of injured forward Ezequiel Lavezzi also made life difficult for Walter Mazzarri's Champions League-chasing side.
In Saturday's other match, Bari inflicted Palermo's first Serie A defeat in eight games with a classy 4-2 home win.
Palermo, in fifth, have been in good form since Delio Rossi took over in November and they dragged themselves level thanks to Edinson Cavani's close range finish and a superb strike from Javier Pastore.
However, the Sicilians then had Fabio Liverani sent off for two bookings and seventh-placed Bari retook the lead through Paulo Barreto's penalty before Vladimir Koman sealed the win with another fine goal five minutes from time.
