The 24-year-old, who has not been named, was held on suspicion of sending malicious communications.
Ffity people were killed and 48 more injured in shootings at two mosques in the city of Christchurch on Friday. A 28-year-old Australian, Brenton Tarrant, has since appeared in court charged with murder.
Referring to the UK arrest, a statement released by Greater Manchester Police said the force "became aware of a post on social media making reference and support for the terrible events in New Zealand."
It added: "Police have made urgent enquiries and a man aged 24 from the Oldham area is now under arrest on suspicion of sending malicious communications.
“This is a very difficult time for people. The events in New Zealand have reverberated around the world. Many people are in deep shock and are worried. It is at times like this that, as a community, we stand together. Where the law permits and people cross the line, we will take robust action, which may include arrest and prosecution."
The action comes as forces across the UK have been attending mosques this weekend to reassure congregations.
Assistant Chief Constable Russ Jackson, of GMP, said: “It is clear that people are worried and we really understand that... It is truly terrible what happened yesterday. It is hard to put into any form of words."
He added: "We have nothing to suggest any threat locally, but none of this can diminish how people feel and that is why we want to be there to offer more support at this difficult time.”
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies