Efe Sodje scored before later seeing red as Bury beat Exeter to reach the FA Cup second round.
The Shakers bossed the opening stages, and took a deserved lead when Sodje rose highest to head home David Worrall's cross at the back post.
Veteran defender Sodje was then sent off for a second bookable offence when he tugged back Alan Gow.
The Grecians almost made the man advantage count, but Gow's narrow miss was the closest they came to levelling.
VIEW FROM THE DRESSING ROOM
Bury manager Kevin Blackwell told BBC Radio Manchester:
"I thought we dominated the first half. We were bright, inventive and should have finished the game off.
"In the second half, they changed. We knew it was a cup game and things will change.
"We lost Efe, who was sent off, but we had a lot of youngsters there that put in a shift.
"Sometimes it's pretty and the sun's shining, and sometimes it's a cold, bleak day and you have to roll your sleeves up. I thought the lads did that very well today when they had to."
Exeter City manager Paul Tisdale told BBC Radio Devon:
"How we didn't get a goal I don't know, I'm very disappointed.
"It's hard to pick fault with anything we did in the second half - they played with a lot of freedom and a lot of fight.
"I think we have to do what everyone else does, I think we should have gone over a few times."