Gravestones in Barnhill Cemetery were damaged in an apparently deliberate fire on Sunday evening.
Several headstones cracked and were left with scorch marks in the blaze which broke out at around 8.30pm.
The fire is thought to have started in shrubs and bushes situated between several stones, and it was extinguished before it spread further.
Two firefighting units attended to put out the blaze.
Councillor Laurie Bidwell, who represents the area, said he was “appalled”.
“The relatives who have loved ones buried in the cemetery will be very upset by what has happened,” he said.
“I would hope that the police will keep a close eye on the cemetery to make sure we do not have some copycat action.
Pretty big fire at Barnhill cemetery. Seems to be burning itself out but could have been much worse. pic.twitter.com/HiJ3XMd9a3
— Stuart Burns (@stuart621) August 14, 2016
“It is very disappointing that this happened in this community.
“There have been reports recently that people exercising their dogs are not cleaning up in some of our cemeteries. It’s disappointing when members of the public, whatever their age, do things which do not give the proper respect these areas deserve.
“I hope the environment department can pretty speedily do some work on the areas that are damaged.”
In June, there were reports of graffiti on the cemetery’s walls.
A Dundee City Council spokesman said: “Pampas grass and a conifer tree were destroyed and there was smoke damage to four headstones.”
They added the site would be cleaned and tidied later on Monday.
