Potter, 32, took third after a remarkable bunched finish, with France's Cassandre Beaugrand winning gold after opening up a gap in the late stages of the run from Switzerland's Julie Derron - who won silver - and Team GB's Potter.
All three top finishers crossed the line within 15 seconds of each other, with France's Emma Lombardi missing out narrowly on the podium.
Beaugrand, who trains in the UK, revealed she had vomited shortly before the start of the triathlon.
"It was the first time it's ever happened to me," she told France 2 television, adding that Britain's Georgia Taylor-Brown had encouraged her to victory, saying: "It's just a race, you can do it."
She said she had been confident in her sprint finish but had to stay alert during the cycling when a number of competitors crashed due to slippery conditions.
World Triathlon and the Paris Games organising committee announced around 04:00 am local time (03:00 BST) on Wednesday that the women's race would go ahead after water tests revealed that bacterial pollution had fallen to acceptable levels in the Seine.
Rain fell overnight in Paris, but had stopped by the time athletes jumped off a pontoon laid over the river at the historic Alexandre III bridge in the heart of the city.
It made the race through the heart of Paris hazardous, with many competitors falling on the wet cobbles during the cycling.
