Simone Pepe had a goal ruled out for a debatable offside decision after also hitting the post, while Mirko Vucinic also struck the woodwork twice as Juve sought to get back to winning ways.
A frustrating afternoon might have been even worse for Juve as well, with Genoa also having several good chances to take the points in an exciting, open encounter at the Stadio Luigi Ferraris.
Rodrigo Palacio had the opportunity to put the hosts ahead after just 20 seconds when he broke through and tried to lift the ball over Gianluigi Buffon, who managed to stretch and push the ball behind with his fingertips.
It was a sign of things to come in a game full of chances, but lacking goals.
In the third minute, Pepe's cross from the right was met by Emanuele Giaccherini at the near post, and he beat Sebastien Frey with his shot, but the ball went inches wide.
Four minutes later, Frey made a good save from Pepe before Claudio Marchisio tested the Genoa goalkeeper's reactions from close range.
Alberto Gilardino headed Genoa's next chance over the crossbar in the 26th minute before Vucinic hit the woodwork twice.
His first effort was a glancing header at the near post which struck the crossbar with Frey beaten before, five minutes later, he headed against the outside of the post.
Pepe should have put Juve in front on the hour mark, but he hit the post when it seemed easier to score from Andrea Pirlo's free-kick.
Kakha Kaladze tested Buffon with a volley at the other end before Marchisio gave Frey more work to do with a shot from 25 yards.
Pepe finally found the back of the net in the 71st minute, but the goal was controversially ruled out for an offside decision.
Gilardino wasted Genoa's best chance of the game in the 79th minute when he beat the offside trap and touched the ball past Buffon, but he cut his shot wide from a tight angle as the match finished goalless.