Napoli, with just one point to their name from the last six on offer prior to tonight, got the job done through Albiol's 72nd-minute header.
Without a win in this fixture since January 2012, Genoa made a positive start and gave their opponents a scare when Darko Lazovic picked out the side-netting with his shot.
The post once again came to Genoa's rescue early in the second period, with Mertens this time rueing his luck after getting a shot in from the edge of the box.
Goran Pandev and Mertens were the next players to pass up decent openings but, just when the match looked destined to be heading towards a goalless conclusion, Albiol diverted Jose Callejon's corner home at the near post.
Only a goal-line clearance from Davide Biraschi prevented Insigne from doubling his side's tally, but the Partenopei held on at the other end to inflict a third successive defeat on their opponents.
NAPOLI (4-3-3): Reina; Hysaj, Albiol, Koulibaly, Mario Rui; Allan, Jorginho, Hamsik (Zielinski 20'); Callejon (Rog 85'), Mertens (Milik 81'), Insigne
GENOA (3-5-2): Perin; Biraschi, Spolli, Zukanovic; Rosi (Rossi 78'), Lazovic (Taarabt 61'), Bertolacci, Hiljemark, Laxalt; Pandev (Rigoni 67'), Galabinov