Morgan had appeared to be out of contention when falling on his first jump in the final, but the Southampton athlete responded with scores of 82.5 and 85.5 to earn third place.
The performance was not without risk, however, with Morgan pulling off a front-side 1440 triple with mute and tail-grab for the first time in competition, and the Briton admitted that he was shocked to have finished in third spot behind Canadian Sebastien Toutant and American Kyle Mack.
The 28-year-old told BBC Sport: "I can't believe it. I didn't think I'd make the podium and I was just stoked to make the final.
"I just wanted to put a show down and I was worried after the first jump that I wasn't going to land anything but managed to land my next two jumps and walked away with a medal."
Morgan's medal ensured that Great Britain reached a total of five for the Games, eclipsing their previous record of four which was achieved in 1924 and 2014.