Cuadrado got Juve on its way in the ninth minute of Monday's (AEDT) game, the wing-back whipping in straight from a corner for his third league goal of the campaign.
Salvatore Sirigu made a string of saves to deny Juve after that, but Dybala ensured there was to be no tense finale with a fine finish eight minutes from full-time.
Despite now winning four of its last five Serie A matches, Juve is still seven points adrift of fourth-placed Atalanta, while Genoa remains 18th and two points from safety.
Juve went ahead when Cuadrado's corner from the left looped over Sirigu and in off the underside of the crossbar, the Colombia international becoming only the eighth player to score directly from a corner in Serie A since 2010.
Sirigu blocked a Federico Bernardeschi effort at his near post soon after, and the Genoa goalkeeper did superbly to keep out Matthijs de Ligt's initial header and close-range follow-up in the 37th minute.
Sirigu was called into action again on the stroke of half-time, this time repelling Alvaro Morata's strike on the stretch after an incisive break.
Dybala whipped over at the start of the second period, before an inspired Sirigu denied Morata, Dybala and Cuadrado in quick succession.
Argentina international Dybala did eventually get the better of Sirigu, firing across him from the left-hand side of the penalty area after being played in by Bernardeschi.