They needed a dramatic penalty shoot-out to do it but the Shakers deserve to be in the hat for the second round draw following a vastly improved performance.
They defended well throughout the earlier 90 minutes with higher league Walsall rarely threatening their goal.
But Jordan Robertson, Andy Bishop and the impressive Simon Johnson were all guilty of missing chances.
However, Bury didn’t give keeper Rene Gilmartin a sniff in the shoot-out with man of the match Brian Barry-Murphy, Richie Baker, Bishop, Ryan Lowe and match-settler Stephen Dawson all netting with style after Sam Parkin had hit the post with Walsall’s first penalty.
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