Fabio Quagliarella put Juve in front in the 54th minute and they appeared comfortable with their lead, until Marco Borriello equalised in the 68th minute.
Juve threw everything forward in search of a winner and had two penalty appeals turned down as tempers flared at the end.
With finding the net in mind, reports on Saturday night claimed former France striker Nicolas Anelka is on the brink of signing for the Turin club.
The only person keeping his cool was Genoa coach Davide Ballardini, who celebrated his return to the Genoa bench with a point which halt a run of two consecutive defeats.
There were only a matter of inches separating Juventus from taking the lead three twice in the first half, and then again just after the interval.
Claudio Marchisio was first to go close in the 22nd minute, but he shot wide at full stretch, unable to get the right direction on his shot.
A minute before half time, there was a repeat when Quagliarella could not find the target as he lacked an inch to make a more telling touch to the ball.
And in the 51st minute, Martin Caceres was the same distance away from getting enough of a touch on the ball in front of Sebastien Frey, who was able to push the ball away.
The home team finally gained the lead in the 54th minute.
A goal was in the air and Stephan Lichtsteiner provided the ball which Quagliarella shot towards goal, and Andreas Granqvist could only help it on its way in.
Juve failed to build on their lead, though, and they were punished in the 68th minute when their former striker Borriello reached Juraj Kucka's cross at the far post to head Genoa level with their first attempt on goal of the match.
Genoa were reduced to 10 men five minutes later when Antonio Floro Flores limped off the field with an injury, but all three substitutions had already been made.
There was nothing else the Rossoblu could do but get all 10 men behind the ball and try to defend the remaining 15 minutes.
The Bianconeri threw everything they could at them, and they had a strong appeal for a penalty waved away when Mirko Vucinic had his shirt tugged inside the penalty area.
Sebastian Giovinco curled a free-kick against the left-hand post, but Juve simply could not find a way through.
With 20 seconds of the game remaining, Juve had another penalty appeal turned down by the referee when Granqvist handled involuntarily in the area, prompting furious scenes with Juve coach Antonio Conte racing on to the field to confront the referee at the final whistle, but Genoa had the point they were looking for.