In the big air event, Canada's Sebastien Toutant won gold with a score of 174.25 while United States' Kyle Mack took silver.
Morgan fell on his first jump but responded with 82.50 on run two and 85.50 on his final run - after landing a front-side 1440 triple with mute and tail-grab, which he had never succeeded in doing in competition before.
He becomes the first British man to win a Winter Olympic medal on snow and only the third Briton after Atkin and Jenny Jones.
"I can't believe it. I didn't think I'd make the podium and I was just stoked to make the final," he told BBC Sport.
"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, from Southampton, finished just 0.75 off silver and his medal was confirmed when Canadian favourite Max Parrot crashed on his final jump.
Morgan had been struggling with a knee injury this season and Matt McCormick had been brought to the Olympics as back-up had he not been able to compete in slopestyle and big air.
"I'm 28 years old and I've maxed the limit of what my body can be scared about," he said.
"It's the best feeling, the pay-off after you do something you are scared about and you land on your feet, that's why we do this.
"For that last trick I pushed all the fear to one side, I was like, even if I completely wreck myself it doesn't matter, I'm just going to go and do it. "
The bronze also sees Great Britain reach UK Sport's pre-Games minimum target of five medals.
It is Team GB's second snowboarding Winter Olympics medal, following bronze for Jones in Sochi in 2014.
UK Sport chair Dame Katherine Grainger said she was "delighted" the British team had reached the medal target.
"I hope that everyone watching at home will be inspired by what they have seen over the past couple of weeks," she added.
GB curlers miss out on bronze
Eve Muirhead's team headed into the women's curling bronze-medal match looking to earn a second successive Olympic medal after finishing third in Sochi four years ago.
But, having lost to Sweden in Friday's semi-final, they were beaten 5-3 by Japan in what proved a dramatic finish after tense and nervy nine ends in Gangneung.
Down 4-3 with the last shot of the match, skip Muirhead misjudged an attempt that would have given Britain two scores and the bronze medal.
"I am absolutely devastated," the Scot told BBC Sport. "We went out to get that end and, as a skip, it's hard when you miss the last shot."
Morgan was the first snowboarder to perform a triple backside rodeo trick, involving three mid-air rotations, and to land a backside 1,800 quadruple cork - four flips and five full rotations - but his best major result was a bronze at the 2014 European X Games.
He finished 10th at the Winter Olympics in Sochi but had his right knee reconstructed two years later and struggled with a left knee injury last year which hampered his training in the run-up to Pyeongchang.
BBC Sport snowboard commentator Ed Leigh
In exactly the same way Jenny Jones worked so hard and was able to crown an impressive career with the glory of an Olympic medal, Billy Morgan has done the same.
He's pioneered so many tricks but has struggled for that core acceptance from the sport. He's come here and landed two breathtaking tricks to win a bronze medal. He has defied the odds.
Other news on day 15
Norway have ran out of commemorative, gold-coloured shoes after winning 38 medals in Pyeongchang, 13 of them gold. "Our ambition was 30 medals in total, but no number was set for gold medals. We will anyhow do our best to provide our athletes with shoes if they win more," said chief communications officer Nils Roine.
A Canadian athlete, his wife and manager have been charged with stealing a car at the Winter Olympics, South Korean police have said.The trio allegedly got into an unlocked car which had been left unoccupied with the engine running in the early hours of Saturday morning in Pyeongchang.
Russian bobsleigh pilot Nadezhda Sergeeva has been banned from the Games for doping - just a day before Olympic chiefs meet to discuss lifting sanctions on her country.