In a close final, the team of Dawson, Josh Bugajski ,Tom George, Mo Sbihi, Charles Elwes, Oliver Wynne-Griffith, James Rudkin, Tom Ford and Henry Fieldman finished behind New Zealand and Germany.
The Germans just edged them out by just 0.2 of a second for silver.
Former Plymstock School pupil Dawson was one of seven Olympic debutants in the boat.
Sbihi, who carried the GB flag at the opening ceremony, was the only member of the crew who had competed at a previous Games.
“We are a talented young crew,” said Sbihi. “I feel like this week we haven’t been able to show ourselves properly, for one reason or another, and part of learning how to win is extracting a performance out of yourself.
“I feel like we did that today. We put ourselves in a position to challenge for the right medal, and on the risk of that you might end up losing a medal.
“I felt like we were able to compose ourselves through the whole piece. These moments are not guaranteed, the guys will try to be there in Paris and we have to take these opportunities when they come.
“Winning is hard. When you’ve won, to keep that run going is even harder sometimes.”
Dawson came through Plymouth Amateur Rowing Club, as did Chloe Brew, who also competed at the Tokyo Games.
Brew was in the Team GB women’s eights that reached the semi-final stage, but which missed out on a place in the A final.
— British Rowing (@BritishRowing) July 30, 2021