The councillors also said the tents were slashed overnight and police are investigating.
Middleton Labour said three people had been staying in tents at the cemetery, including two people described by the group as white British.
In an update published on social media, the councillors said police attended the site following a multi-agency meeting.
They said one man was arrested for immigration offences and detained by police before being referred to Immigration Enforcement.
The group said the other two people had been offered temporary accommodation but declined it.
Rochdale Council's rough sleepers outreach team had been working with the people for several weeks and would continue to offer support, according to the post.
Middleton Labour said all three people had now moved from the cemetery.
The council's Environmental Management Team was also sent to the site to remove rubbish, litter, fly-tipping, damaged bins and abandoned tents. Strimming work had also started.
The councillors said all three tents had been slashed overnight after posts circulated on social media describing the site as a "migrant camp".
They said police were looking for the person responsible and appealed for anyone with information to contact Greater Manchester Police on 101.
Middleton Labour also said police were monitoring social media after comments threatening violence were posted online.
People who experience hate crime or racial abuse were urged to report it to police.
Crimes can be reported to Greater Manchester Police on 101 or online, with 999 used in an emergency.
Rochdale Council's Community Safety Team can be contacted on 0300 303 8878 and its homelessness team on 0300 303 8875.
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