North Wales Police said officers were trying to intervene as the man, in his 30s, attempted suicide in Ruabon Road, Johnstown, at about 10:20 GMT.
Supt Alex Goss said the man died in hospital despite efforts to save him. The officer's injuries were superficial.
The incident has been referred to the Independent Police Complaints Commission (IPCC).
The watchdog said its investigators had been told the man in his 30s sustained self-inflicted knife injuries.
IPCC commissioner Jan Williams said: "Our thoughts are with the family of the man who died following this tragic incident.
"We are in the initial stages of carrying out an investigation into the circumstances and the police interaction with the man.
"We have met with his relatives to explain our role and we will keep them updated throughout our inquiries."
A man has died and a police officer has been injured in an incident in Wrexham.