Napoli, who face a Champions League last-16 first-leg clash at Real Madrid on Wednesday, have 51 points, six behind Juventus and one ahead of AS Roma before their weekend games.
The Naples club, fresh from a 7-1 thrashing of Bologna away last weekend, took the game to Genoa in the first half with tricky forward Dries Mertens pulling the strings.
Kalidou Koulibaly attempted a spectacular flying volley after 33 minutes that flew wide while Lorenzo Insigne had a right-foot shot tipped over by Genoa keeper Eugenio Lamanna.
Just before halftime, Insigne ran on to a Dries Mertens through pass but could only prod the ball wide under pressure and Lamanna smothered Giaccherini's effort.
Napoli went ahead five minutes after the break through Poland midfielder Zielinski who scored with a sweet left-foot strike from just inside the area after a poor clearance.
Mertens soon weaved more magic down the right and made the space to fire a fierce shot at Lamanna who did well to block the Belgian's effort and Amadou Diawara fired the rebound wide.
Giaccherini got the second goal - his first in the league for Napoli since joining from Sunderland - after 68 minutes when Mertens set him up following a mazy burst down the left.
Juventus visit lowly Cagliari on Sunday while Roma travel to second-bottom Crotone. Fourth-placed Lazio, who are seven points behind Roma, host AC Milan on Monday.
(Writing by Ken Ferris,; editing by Ed Osmond)
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