Britain were looking to emulate their female team-mates by going all the way and winning the tournament but fell short in dramatic style, going down 3-2 after extra-time.
Two-goal hero Michael Rae forced the additional period when he levelled the score at 2-2 in the third minute of injury time at the end of the regulation 90.
But after Jake Parrott was dismissed for a second bookable offence in the first half of extra-time, British hopes were dashed when Fanch Weyders struck the winner with just seven minutes remaining.
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France had taken an early lead through Florian Sotoca before a Jack Winter own goal just after the break put them two up.
Rae pounced on 68 minutes to begin Britain's fightback before he added his second right at the death, only for his efforts to be in vain.
Earlier in the day, Japan claimed bronze after beating hosts Russia 3-0.
The British women's team secured gold a day previously with a 6-2 victory over Mexico.
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