The quartet of Harry Leask, Angus Groom, Tom Barras and Jack Beaumont held their nerve during a frantic finish to take second place behind gold medallists the Netherlands.
Australia claimed bronze at Sea Forest Waterway, ahead of Poland, Italy and Estonia.
The medal came after a frustrating morning on day five for Great Britain, which saw the men’s coxless four fail to medal in their flagship event, ending a run of five successive Olympic titles.
There were also fourth-placed finishes for GB in the men’s double sculls and the women’s coxless four.
“It’s a first Olympic medal in this boat class for our country,” a beaming Beaumont told the BBC afterwards.
“We’re standing on the shoulders of loads of people before us, our old team-mates. They’ve helped us get here.
“It’s another step on our way to get a gold.”
He added: “It was wild out there. These conditions are rough, the tailwind, but it’s what we’re used to back at home.
“We’ve trained in this so many times, it did not shake us. We decided, we’re in lane one, outside chance of a medal, let’s take it to them. And we did it. I’m so happy.”