Fabio Quagliarella put the home side ahead with a deflected finish on 54 minutes, only for his strike to be cancelled out 14 minutes later by a former team-mate, Marco Borriello.
Genoa had a penalty appeal of their own rejected early in the match, while Juve felt even more aggrieved when Andreas Granqvist clearly handled in the box in the third minute of injury time.
But the referee signalled a corner, with Bianconeri coach Antonio Conte joining his players in surrounding the officials at the final whistle.
The result means Juventus move six points clear atop Serie A, although Napoli visit Parma on Sunday and can cut the gap to three points.
Struggling Genoa remain third bottom on 18 points from 22 matches.
Quagliarella was on hand to put Juve ahead from Stephan Lichtsteiner's low cross on 54 minutes after the Swiss was expertly found by Mirko Vucinic.
Genoa were second best all night but struck 14 minutes later on a rare foray forward, Juraj Kucka's great work on the right ending with a superb cross from which Borriello gleefully headed home his sixth goal of the season.
The sometime Italy forward refused to celebrate, although his team-mates could not hide their joy.
Despite putting Genoa under intense pressure in a dramatic finish, Juventus could not get the breakthrough, giving Napoli a chance to close the gap at the top.