The athletic duo looked undaunted against the 15-times grand slam champions and converted their fifth championship point to take the match in three hours and six minutes.
Canadian Pospisil and American Sock become the first pair to claim a major tournament at the first time of asking since Lleyton Hewitt and Max Mirnyi took the U.S Open title in 2000.
The 21-year-old Sock has already captured a mixed doubles title at the U.S Open, with Melanie Oudin in 2011.
"To be my second grand slam title and to do it here at Wimbledon is unbelievable," Sock said.
"Our games just clicked perfectly together," Pospisil, 24, said. "We were complementing each other the whole tournament. We were a dangerous team."
The 36-year-old identical Bryan twins, who have won a major tournament every year since 2005, had been hoping to claim a 99th tour victory and a 16th grand slam title.
"These losses are tough," Bob Bryan said. "These are daggers but we usually bounce back."
"It helps to have a few Wimbledons in the trophy cabinet," Mike Bryan added. The twins have won the Wimbledon title three times.
(Reporting By Michael Hann; editing by Clare Lovell)
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