Both defences were holding firm inside the opening 20 minutes, with crosses from both sides being comfortably dealt with.
Bury's Dominic Poleon tried his luck from outside the box on 21 and then 23 minutes, but he was unable to find the target.
Poleon then set up Zac Thompson from the edge of the box and his effort found the corner to give the hosts a 1-0 lead.
At the start of the second half, Liam Noble tried to equalise for Carlisle but his shot on 48 minutes from some distance went wide.
Then 10 minutes later Noble got in a good block as Thompson looked for his second.
It was the home side doing all the pressing and Steven Schumacher tested Mark Gillespie but the goalkeeper saved well.
Peter Hartley could have equalised for Hartlepool in the 82nd minute, but his effort was just over the bar.
Then Garner broke home hearts by firing into the bottom corner in stoppage time.