Cristiano Ronaldo was only named on the bench as one of nine changes by Andrea Pirlo from the 3-1 win over Sassuolo this past weekend.
The hosts got off to the perfect start when Dejan Kulusevski opened the scoring with his fourth league goal of the season after two minutes.
The hosts had to wait until the 23rd minute for the for the second goal as Kulusevski turned provider to pick out Alvaro Morata who found the bottom corner with a neat finish.
Genoa did pull a goal back five minutes later as an umarked Lennart Czyborra sent a fine diving header past Gianluigi Buffon from Edoardo Goldaniga's cross.
Kulusevski almost restored the home side's two-goal advantage before the break but his powerful strike was kept out by Genoa shot-stopper Alberto Paleari.
The visitors were nearly back on level terms 10 minutes into the second half when Marko Pjaca rattled the crossbar with a dipping effort from 25 yards.
Even with the introduction of Leonardo Bonucci, Genoa managed to bag the equaliser on 74 minutes courtesy of a superb long range effort from Filippo Melegoni that sailed into the top corner.
Ronaldo was eventually introduced with two minutes left of regulation time as Wesley dropped to the bench, but it didn't make any immediate impact as the game went to extra-time.
Juventus went ahead in the 104th minute when substitute Hamza Rafia poked home from close range after some good work by Morata.
That proved to be the winning goal as Pirlo charges progress to the last eight while Genoa can now focus on Serie A.