The £10,000 award is the first of a new drama category in Britain's oldest literary awards. It was launched last year by the University of Edinburgh.
Tim Price said: "I am thrilled to be recognised by such a prestigious award, and honoured to be chosen as winner from such a fantastic shortlist of plays - that it is the first time a category for drama has been added to the James Tait Black Prizes is even more exciting."
The play is currently being performed at Edinburgh Festival Fringe.
A verdict of accidental death has been recorded for a Pembrokeshire man who went missing after going to a Six Nations match at the Millennium Stadium, and was found in the River Ely in Cardiff more than a month later.
Ben Thompson, who was 34 and from Haverfordwest, was last seen in Cardiff city centre on the evening of Saturday 2 February.
The inquest heard he then tried to walk back to his hotel, after a day of drinking, but fell into the river.
His body was found on Tuesday 12 March, after an enormous search involving friends and family.
South Wales Police have confirmed that the body found in the River Ely on Tuesday is that of 34-year-old Ben Thompson from Haverforwest.
The identity of the body was confirmed today, and police say there are no suspicious circumstances.
Ben Thompson was reported missing on February 2nd, after he'd been to the Wales v Ireland Six Nations game at the Millennium Stadium.
A search operation involving up to 50 officers, including detectives and specialist search teams, spent many hours searching the Leckwith area of Cardiff where he was last seen.
Chief Inspector Eddie Ough, from South Wales Police, said: “At times like this the people of Cardiff really come together and South Wales Police would like to thank local residents, businesses and the media for their co-operation during our investigation.
“We wanted to find Ben alive and well but sadly that was not to be. We hope now at least that his friends and family have an element of closure and can be allowed to grieve in peace.”
South Wales Police says a body has been found in the River Ely, near Penarth Road in Cardiff.
Formal identification has not yet taken place.
The family of 34-year-old Ben Thompson from Haverfordwest, who has been missing for more than a month after watching Wales play in the Six Nations, has been informed and are being kept updated by family liaison officers.