The place had for decades been known as Green's Playhouse, but from then on it would be known as the Apollo.
In that long-ago era, when Ted Heath was in 10 Downing Street and President Nixon's Watergate-plagued resignation was still 11 months away, the Apollo quickly established itself as Scotland's - and indeed one of the UK's - premier pop and rock venues, alongside its self-proclaimed status as Europe's largest luxury cinema.
In Cash's wake came the Rolling Stones, who played two nights, and Lou Reed, the Moody Blues, Status Quo, Diana Ross, Genesis, Don McLean and Wishbone Ash.
Most of these artists are still going today. The Apollo itself, however, lasted only 12 years - until 1985, the year of Live Aid and the opening of the SECC. But those 12 years were frequently glorious.
Mature student Kenny Forbes, a lecturer in commercial music at the University of the West of Scotland, who is doing a PhD on the Apollo at Glasgow University, traces the venue's legendary status to its first four years.
"It had an amazing run of bands that were at their commercial peak," he said. "Because of the constraints in Britain's concert network at that time, they had to play places like the Apollo. The venue opened at the perfect time."
The place was cavernous and unglamorous but the atmosphere it generated was little short of electrifying. Many artists spoke of the fearsome reputation of the venue and its audiences, but in the end they would be won over, and it was not uncommon to hear them describe their Glasgow appearance as one of the best concerts of their careers.
In 1978, when the then owner's lease expired, leisure giant Mecca caused outrage with plans to turn the Apollo into a bingo hall, but a potent public campaign ensured it survived intact. But by 1985, when it was in a state of considerable disrepair, the Apollo closed, the final concert being Paul Weller's Style Council.
Today, the space is occupied by a skyscraper cinema and the Walkabout pub, but the Apollo's name lives on. Books have been written about it, a musical has been staged, and a website remains highly popular.
Mr Forbes added: "Why does it still resonate with so many people nearly three decades after its final show? It seems to be a generational thing. A lot of people went there while in their early teens. Seeing, at the age of 13 or 14, legendary bands of the stature of The Who and Rolling Stones is the sort of thing that tends to leave a mark. The Apollo attracted many people from west Scotland and further afield.
"Many fans, even now, recall hoping the band would finish its encore so they could make a mad dash for the last train home. The journey to and from the Apollo could be an adventure in itself.
"The Apollo was very much of its time. Urban planning was starting to make a real impact on the city but for many the Apollo was the last remnant of a bygone age. The Apollo, in a sense, characterised Glasgow as it was at that time."
l Anyone with memories of the Apollo can contact Mr Forbes at firstname.lastname@example.org