It is a third triathlon medal for Team GB at these Games after individual gold for Yee and bronze for Potter.
Great Britain were defending champions in the event, which sees each athlete complete a 300m swim, 7km cycle and a 1.8km run.
There had been concerns the event would not go ahead because of poor water quality in the Seine, which has been a running thread throughout the Games.
However, it got under way on Monday morning as planned, despite a request from teams to delay the competition to allow for more athlete preparation time after the familiarisation swims were cancelled.
On a beautiful morning, Yee gave Britain a superb start, roared on by the packed crowds that lined the bridges and Parisian streets.
An excellent bike leg and then a trademark strong sprint saw Yee give Britain an early lead, before Taylor-Brown produced an outstanding performance on the bike.
But she was caught on the line by Germany’s Lisa Tertsch to trail by one second going into the third leg.
Lasse Luehrs kept on the pressure, running right on the heels of Britain’s Dickinson on the final stretch.
Dickinson withstood the pressure and tagged in Potter with a lead.
But the thrilling finish saw Potter, Lindemann and Knibb jostling for first, with Potter at one point finding herself in third.
The trio sprinted for the finish line, with Potter and Knibb barely separated in the photo.
While it is not the silver Team GB initially thought they had won, they will leave Paris with a medal in every triathlon event, which will be of some comfort.
“I really don’t think Beth Potter could have given any more,” former British triathlete Annie Potter said on BBC TV.
“I just think the fatigue from winning the bronze medal in the individual was too much for her.
“You could see how fatigued she was when she crossed the line.”