Halftime substitute Musa Barrow and Timothy Castagne scored early in the second half to lift Atalanta to third, despite having two first-half goals disallowed for offside following VAR reviews.
Atalanta have 65 points, two ahead of Inter Milan, who visit relegated Chievo on Monday. AC Milan, away to Fiorentina on Saturday, and AS Roma, who host Juventus on Saturday, both have 59. The top four qualify for next season's Champions League.
"Today, we couldn't afford to slip up and we did really well," said Atalanta coach Gian Piero Gasperini. "We want to fulfil a dream for the fans whose enthusiasm has been incredible.
"They left huge spaces for us to finish the game off but instead we decided to sit back, and we ran a huge risk, but it all ended well."
Playing at Sassuolo's Maipei stadium because rebuilding work on their own stadium is under way, Atalanta thought they had gone ahead after nine minutes with a Robin Gosens goal, but it was disallowed for offside against Duvan Zapata.
Zapata had the ball in the net himself shortly afterwards but was again offside.
Despite missing playmaker Alejandro Gomez, Atalanta broke the deadlock one minute after the restart as Marten de Roon released Barrow with a pass through the centre of the Genoa defence and the Gambian fired past Ionut Radu.
Castagne turned in the second seven minutes later although Atlanta had to survive a nervous finale after Goran Pandev pulled one back in the 89th minute for Genoa, who are 16th and four points clear of the relegation zone.
(Writing by Brian Homewood; Editing by Christian Radnedge)
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