Crewe halved the deficit through Harry Davis' penalty, but never looked like preventing Bury's passage into the second round.
"It's a bad night. No disrespect to Bury, but we should be coming here and comfortably winning the game," said Davis.
"After 15 minutes I thought we were going to win the game. We started well, caused them a lot of problems and missed two or three chances as well as scoring.
"We could have wrapped the game up and been two or three-up, but that wasn't to be and they got back into the game because of our ill-discipline.
"It's hard to find positives. The first quarter of an hour was a good start, but after that it was just bitty.
"We had periods of good play, but that was it. There weren't many positives and you can't find many when you come to Bury and lose 3-2.
"We've scored five goals this season, but conceded six and five of those were from set-plays. That's a concern and we have to do better."
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