Page topped the standings with two routines to play, but was overtaken by Chinese Zhiu Xueying and Liu Lingling.
This is a second Olympic medal for Page, who won silver at Rio 2016.
His medal was won after Team GB rowers clinched a bronze in the men’s eight.
The team – Josh Bugajski, Jacob Dawson, Tom George, Mohamed Sbihi, Charles Elwes, Oliver Wynne-Griffith, James Rudkin, Henry Fieldman and Tom Ford – were not considered the medal favorites before the race and finished behind New Zealand and Germany.
However, the medal is Britain’s second in the Tokyo Olympics rowing regatta after a silver medal in the men’s quadruple on Wednesday.
Earlier, the GB team enjoyed success in the pool, with medals for swimmers Duncan Scott and Luke Greenbank.
Scott won silver in the 200m individual medley and Greenbank won bronze in the men’s 200m backstroke.
The 24-year-old already had a gold medal in the men’s 4x200m freestyle relay and a silver medal in the individual discipline.
He was fifth before the final 50m of the IM on Friday morning, but found another speed to move up to second, finishing just 0.28 seconds behind China’s Shun Wang.
Scott told the BBC: “It was always going to be tight, it was always going to be exciting.
“I am really happy with the swimming. It wasn’t quite there. “
Greenbank had qualified second fastest in the morning final and started well, just behind Evgeny Rylov of the Russian Olympic Committee after the first pitch.
Halfway through, Greenbank had been passed by American Ryan Murphy, but he managed to hang on to third place with a time of one minute and 54.72 seconds – just 1.45 seconds behind the medalist. ‘or Rylov.
Greenbank, 23, told the BBC: “It’s amazing. A dream come true. I am so happy and over the moon.
“My first memory of swimming is watching the Beijing Olympics.
“It has always been the ambition of my life, so to come away with a medal, an individual medal as well, is absolutely incredible. “
Earlier there was disappointment for GB rower Vicky Thornley, who drastically missed a medal in the women’s single scull at Sea Forest Waterway.
She lost bronze to Austria’s Magdalena Lobnig – and had tried to clinch a second Olympic medal after winning silver in doubles alongside Dame Katherine Grainger in Rio five years ago.
New Zealander Emma Twigg won the gold medal.
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Thornley said: “The fourth is tough, but I don’t think I could have done more, especially in the second half of the race.
“I didn’t have enough at the end, but I’m proud of what I’ve delivered over the past eight days. There were some hellish days, especially the three-day wait for the semi-final.
“The past five years have been amazing – some of my darkest days and some of my best. I am proud of the race I organized.
“I thought I had the momentum, but I didn’t have enough. “
In track and field, Team GB’s Dina Asher-Smith advanced to the 100-meter semi-finals after a time of 11.07 seconds in the first set.
Asher-Smith, who qualified 11th fastest for Saturday’s semi-finals, said: “I have another level – of course yes, it’s the Olympics.
“To be honest today it was just about getting to the next round safely knowing that I have another level to give tomorrow so I’m really happy.
“You try to go and do well. As an athlete you like a full stadium, you like the spectators but at the same time it is what it is and you always have a job to do. “
Team GB’s Daryll Neita set a personal best 10.96 seconds to qualify and Asha Philip also took the next step.
In the 800m, Jemma Reekie advanced to the semi-finals after winning her race in 1 minute 59.97 seconds.
Despite the impressive result, she said: “It was not quite as I would have liked, I did not do what I wanted but I finished and that is the main thing”.
Keely Hodgkinson also managed to qualify, having finished second in his race in 2 minutes 0.159 seconds and Alex Bell finished fourth, reaching the semi-finals as one of the fastest six outside of automatic qualifying.
In the high jump, Briton Tom Gale qualified for the final with a season record of 2.28m.
But the GB team missed out on a place in the discus final after Lawrence Okoye recorded three pitching steps.
Zak Seddon did not progress from his obstacle course to crown a nightmarish Games.
He was one of six track and field athletes from Team GB forced into self-isolation after being reported as close contact from an infected passenger on their flight to Tokyo.
He said: “It hasn’t been great and mentally it’s been difficult here, there and everywhere. These days sport is what it is and some athletes are going to make races easier than others, but you have to face it. ”