Roy Buggins, 51, had 29 years' service as an officer.
He was described as having been a "highly valued and well-respected" member of the Montrose and Brechin Community Policing Team.
Mr Buggins died on duty after falling ill on Tuesday September 3.
Police Scotland said there were no suspicious circumstances surrounding the death.
The funeral was held on Tuesday at Parkgrove Crematorium in Friockheim, Angus.
A bagpiper walked in front of the procession and officers formed a guard of honour as the coffin was led from the hearse to the service.
It was draped with a sheet bearing the Police Scotland logo and colleagues could be seen welling up as it was carried past them.
Police Scotland Chief Constable Iain Livingstone was among those in attendance.