The branch went over a bank, which would have made things difficult for the owner to rescue the animal herself.
The spokesman said one of their crew managed to get to the goat safely and free her.
"He held her by the horns and she landed safely.''
The rescue took about 15 minutes.
"It's just one of those bizarre things,'' the spokesman said.
"We always get called to animal rescues, but that was the first time I'd been called to a goat rescue.''
Asked for his name, the firefighter laughed and asked not to be named as he might get some stick from his colleagues.
"They'd get on my goat if I gave you a name.''