Alex goalkeeper Ben Garratt saved from Leon Clarke and Tom Soares, while Andrew Tutte fired over the bar.
But the visitors improved after the break and Ryan Colclough put his shot inches wide from 20 yards.
Bury continued to create and waste chances but Crewe almost won it in the 90th minute when Marcus Haber's shot was stopped on the line by Reece Brown.
The Alex move off the bottom of the table but are seven points from safety, while Bury move up to 12th and are six points adrift of the play-offs.
Crewe Alexandra remain deep in League One relegation trouble despite a spirited goalless draw at Bury.