Britain's Nick Rogers and Joe Glanfield had to settle for a silver medal after a tactical battle in the final race of the men's 470 class.
Needing to finish four places ahead of Paul Foerster and Kevin Burnham, the British pair were unable to overhaul their American rivals.
Rogers and Glanfield started race 11 already assured of a silver medal.
But the Americans, who finished 23rd, kept them pinned at the back of the fleet, with the Britons in 24th place.
After missed out on a bronze by one point in Sydney four years ago, they were delighted at the outcome despite missing gold.
"We didn't win but we are really happy just to get the silver," Glanfield said.
"When we look at the quality of sailors walking away from here without a medal, we are over the moon.
"If you had asked us what we didn't want it would have been light winds all week, and that is what we got, so to get a siver is a big achievement for us."
The pair picked up Britain's third sailing medal of the Games, after golds for Shirley Robertson's Yngling crew and Ben Ainslie in the Finn.
Foerster and Burnham finished with 71 points, three better than the British pair's tally of 74, while Japan's Kazuto Seki and Kenjiro Todoroki claimed the bronze.
Rogers and Glanfield had led the fleet from the early stages of the event, and were 10 points ahead after a 10th place in Thursday's first race.
But a disappointing 19th in race 10 left them trailing by two points, and needing to finish in the top 15 and at least two boats ahead of the American pair on Saturday.
"We came up with a strategy for the start and Paul (Foerster) started sailing towards us as we expected," Rogers explained.
"But about two minutes before the start the wind really dropped and we couldn't make the move we wanted."
"The big problem was they didn't have to get a result," Glanfield added. "They could put everything into taking us out and did a good job to get the better of us."