Louis Smith, Max Whitlock, Daniel Purvis, Sam Oldham and Kristian Thomas scored a total of 271.711 as they produced a prodigious team performance.
Britain had initially been awarded silver only for an inquiry into the score of Kohei Uchimura's pommel horse routine to allow Japan to claim silver with Britain then downgraded to bronze. China defended their gold medals.
The last time Britain was celebrating an Olympic team medal came during the Stockholm Games in 1912. Their reclassification as bronze winners following a Japanese appeal threatened to take the shine off their accomplishment, but a medal of any colour still exceeded all expectations for a British team seemingly dwarfed by the gymnastic talents of the United States, Russia, China and Japan. But the US faded in a tense finale and Team GB held off a challenge from Ukraine to secure their place on the podium.
Ukraine were in bronze medal position and Japan in silver with the final rotation to go, but brilliant performances on the floor by Whitlock, Purvis and Thomas enabled them to win a historic medal.
"We came into this competition not expecting anything," said Whitlock. "To be honest [the downgrading] doesn't matter. A bronze or silver medal doesn't matter. We've done more than we could have asked for."
Oldham, who was unable to train six months ago due to injury, was similarly thrilled. "I just can't believe it as we just walked away from a home Olympics with a bronze medal. It's unbelievable. We put in so much hard work and we're all so happy."