After the race was given the early-morning green light following pollution tests, Beaugrand was among the leaders from the start and then put the hammer down with a kilometre of the 10km run to take gold, breaking the tape in 1:54:55.
Julie Derron continued Switzerland’s rich pedigree in women’s Olympic triathlon, coming in six seconds behind to add a silver to the golds of Brigitte McMahon in 2000 and Nicola Spirig in 2012.
World champion Potter also built on Britain’s record as the most successful nation in the sport, with her bronze a further nine seconds back making it nine medals for the country in all.
“I threw up before the start, I was in a total panic," said Beaugrand, who failed to finish at the last Olympics after suffering a puncture.
"It was nerves, it’s never happened to me before, and in front of the other athletes. Everyone knew I was stressed. It’s not what you want to do.
"I told myself ‘Cass, you can’t go through the same thing as Tokyo again, concentrate, you’ve been doing triathlon your whole life, it’s just another race.'
“To do this in front of this audience, I can’t believe it.”
After the postponement of the men’s race on Tuesday due to poor water quality, the women's event got underway on time on the back of a rainstorm.
Defending champion Flora Duffy, the winner of Bermuda’s only Olympic gold and appearing in a remarkable fifth Olympics at the age of 36, led the 1,500 metres Seine swim out.
All the main contenders were in a group of nine that picked Duffy up but the morning’s heavy rain, combined with several cobblestone sections, made the bike leg treacherous and there were a series of crashes.
The leaders – with the exception of Germany’s Laura Lindemann - negotiated the seven-lap 40km route safely and, without any hills to break them up, arrived en masse for the 10km run.
After the first of four 2.5km laps, however, Potter, Derron, and local favourites Emma Lombardi and Beaugrand had forged clear.
The quartet ran together until halfway round the final lap when Beaugrand made her decisive move and was able to enjoy the greatest moment of her career.
'Going for gold'
Derron, who was at the front of the breakaway foursome for most of the race, said it was "a bit daunting".
"Running at the front you have no idea what’s going on behind you but I feel more comfortable like this," she said.
"I could run my own stride and my own rhythm, that was just perfect for me to be there. I’m so pleased it paid off.”
Potter, who raced the 10,000 metres on the track at Rio 2016, said: "I'm so happy. I was going for the gold but Cassandre and Julie were just too good for me today.
“It said it was going to be dry but it was obviously very wet. I live and train in Leeds so I am used to those conditions. I didn't want to make any silly mistakes and tried to stay as safe as possible.
"I've come a long way in eight years... I dug deep and found something else."
Lombardi was a frustrated fourth while Duffy, despite being out for a year with a knee injury, took an impressive fifth. Britain’s Tokyo silver medallist Georgia Taylor-Brown finished sixth.
The men's race went off at 10:45 CET.
