Page, 25, was beaten to gold by 0.425 points as Canada's Rosannagh MacLennan won a second successive Olympic title with a score of 56.465.
World champion Li Dan of China took bronze, while Britain's other finalist, Katherine Driscoll, came sixth.
"I can't believe I've won an Olympic medal. I'm just so happy," said a clearly emotional Page.
"I had no idea it was a medal but my performance was the best I could have done, that is why I was so happy, that all the training and hard work had paid off.
"I am so happy I got to share the final with Kat, one person would have been history and we got two in the final."
Prior to winning Olympic silver, Page's highest career finish in an international event was fifth at the 2015 World Championships in Denmark.
Trampoline made its Olympic debut in 2000 and this is the first time British women have reached the final.
Bryony Page became the first British woman to win an Olympic trampoline medal by claiming silver in Rio.