Touts were reportedly selling tickets to the Birmingham Odeon concert at more than 40 times their face value - over Â£100 - such was the demand.
More than 44,000 people wanted tickets to the 2,397-seat venue, its then-manager Tony Harvey recalls.
The band won Eurovision at the Brighton Dome on 6 April 1974 and went on to sell more than 400 million albums.
A stage technician at the Birmingham Odeon remembers rigging a quadraphonic sound system to replicate a helicopter swirling around the auditorium, which has since become a cinema.
Mr Harvey, who managed the venue, brought his daughters Tania and Tracey to watch.
"We were besieged by people wanting tickets," he said.
"From the moment the tickets went on sale I became the most famous person in Birmingham.
"I don't think police expected so many people to turn up."
"I remember being kept in the dark how they were going to arrive. They came out of the fire exit at New Street Station opposite our stage door."
Dale Michelsohn, originally from Bournville, Birmingham, but who now runs a venue in Skansen, Sweden, worked as a stage technician at the Odeon from 1974 to 1988.
He said: "The most unusual thing for us was the use of a quadraphonic sound system, something not often done - I think Pink Floyd had used one earlier.
"It was only for the opening. We had to quickly de-rig it during the start of the concert and drag it through the exit doors at the rear of the stalls," said Mr Michelsohn, who now runs the Birmingham Odeon Memorial group on Facebook.
Rafael Cavar, who wrote he went to the gig on the website Birmingham Music Archive, said: "The demands were so high they could have sold out the Odeon over 10 times.
"I was very lucky to get my ticket on the black market outside the Odeon at a fair price.
"People were paying over Â£100 (40 times the face value), which was a fortune in those days."
Four decades ago today, flying high on their Eurovision success, Abba played their first UK gig.