Media playback is not supported on this device
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 another Briton, Katherine Driscoll, came sixth.
"I can't believe I've won an Olympic medal," said Page, from Huntingdon. "I'm just so happy."
Find out how to get into gymnastics with our special guide.
"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."
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.
Media playback is not supported on this device
Subscribe to the BBC Sport newsletter to get our pick of news, features and video sent to your inbox.

The Iron are currently sixth in the National League, one point and one place outside the play-off spots.
"I thought we asked a lot of questions of a very good Barrow team," Cowley told BBC Essex.
"When you come off the back of a six or seven-hour journey and you concede a goal after five minutes it's always going to be tough."
Barrow extended their unbeaten league run to nine games with the victory, and although Braintree's losing run extended to three games, Cowley remained positive.
"What we've done so far has been nothing short of remarkable," he continued. "But if anyone thought this journey was going to be easy, then they're living in cloud cuckoo land.
"We're a part-time team. This is our toughest challenge to date, but we're going to embrace it. We're going to enjoy it and see where it takes us."

The 26-year-old will join Wednesday on a permanent transfer in the summer for a fee believed to be £10m.
The Owls had to wait until Wednesday for the English Football League to decide if the deal had been completed by Tuesday's 23:00 GMT deadline.
Rhodes scored six goals in 24 league appearances for Boro after joining from Blackburn for £9m in January 2016.
However, the Scotland international, featured just six times this season and failed to score.
He started his career with Ipswich and had loan spells with Oxford, Rochdale and Brentford before joining Huddersfield in July 2009.
Rhodes scored 73 goals in 124 league appearances for the Terriers and helped them to promotion from League One in 2012.
Blackburn paid the Terriers £8m for him in August 2012 following their relegation from the Premier League.
He moved on to Boro last January after scoring 83 goals in 159 league games for Rovers and helped Aitor Karanka's side win promotion to the Premier League.
Rhodes' father, Andy, is the goalkeeping coach at Hillsborough.
Find all the latest football transfers on our dedicated page or visit our Premier League tracker here.
Bryony Page became the first British woman to win an Olympic trampoline medal by claiming silver in Rio.
[NEXT_CONCEPT]
Braintree manager Danny Cowley said the task ahead of his team is "a brilliant challenge" after the defeat at Barrow.
[NEXT_CONCEPT]
Sheffield Wednesday have signed Middlesbrough striker Jordan Rhodes on loan until the end of the season.