The Pacific Islanders overcame New Zealand in the quarter-finals before coasting past Japan to set up the meeting with Team GB, who had impressed in their victory over South Africa in the semis.
Britain's defence had been a particular strength throughout the tournament, but it was obliterated by the Fijians as they all but secured the historic gold by half time.
Ben Ryan's side scored five unanswered tries in the opening period to take a 29-0 lead into the break, and they extended that further after the restart.
GB were no match for the World Series champions and were forced to settle for silver, while South Africa claimed bronze.
Fiji, meanwhile, not only claimed the first sevens gold ever, but also picked up the first Olympic medal in the country's history.