Foad, who won Olympic bronze in the eight at London 2012 last summer, along with newcomer Cox were the last British boat to be named for the World Championships.
And they put in a gutsy row to finish behind Poland but ahead of Australia in second in the men's pair repechage to advance to the semi-finals and secure a place in the world’s top 12.
"We had a good first 1km," said Foad, who claimed successive World Championship silvers as part of Britain's eight in 2010 and 2011.
"I looked across at that point and wanted to clear the race up early and we put in a push and then we probably tied up bit more than expected at the end."
However the women's quadruple scull of Lucinda Gooderham, Monica Relph, Kristina Stiller and Rosamund Bradbury couldn't follow Foad and Cook's lead as they came home fourth in their repechage.
The British quartet couldn't keep up with the pace set by the USA and Poland and missed out however they will still race again in the B final in South Korea.
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