Bury's 5-2 demolition of Conference North outfit Vauxhall Motors was marred by the loss of Dave Challinor.
The skipper was stretchered off a minute into the second half with what looked like a serious knee injury.
But the blow had no impact on the outcome as the home side cruised into the second round for the first time in five attempts.
Two goals from midfielder Dwayne Mattis and strikes from Chris Porter and Challinor put Bury four up at the break.
The visitors reduced the deficit through fine goals from Wayne McDermott and Karl O'Donnell before Dave Nugent completed the scoring for Bury.
Motors' Peter Cumiskey should have opened the scoring on six minutes, but Andy Marriott denied him with a fine save, and from then on the first half was one-way traffic.