Heading into the last of the six rotations, just 0.640 points separated the ROC in first from China in second, with Japan only 0.631 further adrift in third.
China were the first to finish their competition and briefly assumed the gold medal position with a total of 261.894 but Daiki Hashimoto produced a near-flawless routine under pressure on the horizontal bar worth 15.100 to hurl the host nation into the lead.
But the world all-round champion Nikita Nagornyy was still to perform with the last routine of the night on floor and he held his nerve to deliver gold – the Russians’ first in this event since 1996 – by a margin of just 0.103 points.
The British quartet of double Olympic champion Max Whitlock, Joe Fraser, James Hall and Giarnni Regini-Morgan reached the final by qualifying in fifth and improved a place here but were never in serious medal contention, finishing more than six points behind bronze medalists China.
The high-point from a British perspective was the continued good form of 22-year-old Joe Fraser, who qualified fifth for the individual all-round competition later in the week and will also bid for a medal on the parallel bars, where he is the reigning world champion, at his first Olympics.
Whitlock, meanwhile, warmed up for the defence of his pommel horse title with a score of 14.966, the second-best performance of the night on that piece of apparatus.