The 23-year-old from Edinburgh ran a personal best of three minutes 29.05 seconds to finish behind Norway’s Jakob Ingebrigtsen and Kenya’s Timothy Cheruiyot.
Ingebrigtsen overtook Cheruiyot on the final bend and raced clear to win in a new Olympics record of 3min 28.32sec.
Kerr’s team-mates Jake Heyward and Jake Wightman finished ninth and 10th respectively at the Olympic Stadium.
It was a superb run from Kerr who held off the challenge of Kenya’s Abel Kipsang in the home straight and almost overhauled Cheruiyot on the line.
It was Britain’s first men’s 1500m Olympic medal for 33 years and came 24 hours after fellow Scot Laura Muir won silver for Team GB in the women’s 1500m.
“I'm blown away,” said Kerr, who struggled in the qualifying in Tokyo, finishing seventh in his heat and only scraping through as a fastest loser. “This has been a hard Championships for me, the first run wasn't great, it was one of those days and you can have those. Sadly mine was the first round of the Olympics.
“I had to go back, think about it, recalibrate and come back to these next rounds fighting for every single step. I feel like you saw that today, you saw that in the semi-final and I'm really happy with that performance.
“I have this weird confidence in myself. Some may call it cockiness, some may call it general confidence.
“When you put the effort in and you're surrounded by a team like I am, you can't not feel confidence every step of the way. I'm so pleased I can give back to all those people who were able to sacrifice for me.
“When the first medal came back to our Team GB camp through Keely [Hodgkinson in the 800m] there was a sense of enjoyment through someone else.
“I had to take that away and think, 'I want to create that for myself.'”
Kerr took a mammoth 2.5 seconds off his personal best and was also just 0.24sec outside Mo Farah’s British record time over 1500m and the third fastest by a British man in history.
It is a discipline in which Britain has a strong Olympic history, winning five golds, more than any other nation. In addition, fve of the nine podium places during the Olympic Games of the 1980s taken by Brits.
It has been a long wait since those heady days of Seb Coe, Steve Ovett, Steve Cram and Peter Elliott but Scotsman Kerr joined their ranks as he crossed the line in third place.