The 19-year-old from Flintshire set an F20 shot put personal best of 12.94m to secure bronze.
Poland's Ewa Durska set a world record to win gold with Anastasiia Mysnyk of the Ukraine winning silver.
"You can't picture something as amazing as this. I came here for a PB. I didn't think I'd get a medal as well," Fortune said.
"It was an absolute honour to run around that track with the GB flag flying behind you.
"I just hope, maybe in Tokyo, I'll get the gold medal and listen to the anthem played."
Sprinter Rhys Jones ran a personal best to finish fourth in his heat to qualify for Sunday's T37 100m final.
Sara Head will face Chinese world champion Li Qian in the women's singles class 3 table tennis quarter final after a 3-0 last 16 victory over Croatia's Helena Dretar-Karic having earlier beaten Sigala Lopez of Mexico.
"I'm over the moon. I was controlled, composed, kept to my game and that worked. It was so much fun," Head said.
In the men's singles class 1, Rob Davies will play Endre Major from Hungary in Sunday's semi-final following a 3-0 win over Jean-Francois Ducay of France.
Ducay had beaten Davies in the group stages at London 2012 and Davies was relieved to make amends.
"He is one of the top guys in Paralympic table tennis in my class and has been for years and it has been a battle with him for eight years and to get my revenge on him for London is sweet for sure," Davies said.
Fellow Welshman Paul Davies, having beaten Brazil's Aloisio Lima earlier in the day, could not repeat his heroics against Ki-won Nam and the Korean was a 3-0 winner in the men' singles class 1.
Welsh judoka Jack Hodgson lost to Kento Masaki of Japan in the quarter-finals of the men's +100kg category and although he beat Turkey's Dursun Hayran in the repechages the 19-year-old later lost to Ilham Zakiyev of Azerbaijan.
Clare Griffiths and GB's Women's Wheelchair basketball team secured their third win with a 50-45 victory over Germany.
Sabrina Fortune has become the first Welsh athlete to win a Paralympic medal at Rio 2016.