The worker suffered severe blood loss, and was taken off site but could not be resuscitated.
Another man was also rushed to hospital with minor injuries.
The fatal incident occurred just after noon on Thursday on the deck of the bridge’s north tower.
Harry Frew Regional Secretary for UCATT Scotland, said: “UCATT sends its sincere condolences to the family of the worker who has suffered this tragic accident. It is impossible to imagine how they must be feeling.”
The fatality is particularly poignant as it occurred on Workers’ Memorial Day.
It is understood the victim, named locally as John Cousins, had taken part in a minute’s silence for construction victims just before the tragedy.
His colleagues downed tools for the day on Friday in honour of his memory.
Pat Rafferty, Unite’s Scottish Secretary, said: “Our thoughts are with the deceased’s family to whom we offer our profound condolences.
“Once again, a loved one has gone to work and will not return home.
“That is hard to swallow any day but on today of all days, on international Workers’ Memorial Day, when we remember those who have perished at work, this loss is especially poignant.
“We call on the Scottish Health and Safety Executive to thoroughly investigate what has happened here.”