Britain, finished the final on 37 points, were left to rue the ineligibility of team anchor John Whitaker for the final.
John Whitaker had been expected to jump in the final, even though he had withdrawn from Sunday's first round because of a problem with his horse Peppermill.
Britain were laying joint fourth after round one and had high hopes of a medal, but shortly before the start of the competition the International Equestrian Federation (FEI) upheld the objections of seven of the other eight national teams competing for medals.
The FEI's decision put enormous pressure on Britain's three remaining riders, who had to compete without the luxury of a discard score, and after a disappointing opening jump by Nick Skelton a top-three finish appeared remote.
Stockdale, on Corlato, then hit two fences for eight penalty points to leave Ben Maher with little to jump for. The 25-year-old produced a magnificent clear final jump on Rolette to move Britain ahead of eighth-placed Sweden and ninth-placed Australia.
Skelton, Stockdale and Maher will compete in the individual show jumping final on Thursday.
The United States won gold in the team show jumping with three clear rounds in a dramatic jump-off with Canada.
Canada, who had finished equal on 20 points with the defending champions after the allocated jumps, had to settle for silver in Hong Kong.
Norway secured bronze with 27 points, their best score at an Olympics.
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