Cox qualified a place for the round of 16 but then narrowly missed the cut in the ski off for the final six.
Nicole Parks had the best day of her career, qualifying in 10th place for the first final and then finishing in 11th.
In the men's event, Dale Begg-Smith was the best placed of the Australian's finishing 12th, with Matt Graham back in 13th. The two had exactly the same score, but were separated on a countback.
The women's world cup was won by Canadian Justine Dufour-Lapointe, with US skier Hannah Kearney and Chloe Dufour-Lapointe in third.
Canadian Mikael Kingsbury produced a hometown win in the men's, taking the title from his countryman Alex Bilodeau with Partick Deneen from the USA rounding out the podium placings.
Other Australian results in the 1988 Olympic Winter Games host city saw Taylah O’Neill finishing 22nd in the women’s competition, while Brodie Summers was 29th in the 64-strong men’s field and Sam Hall 40th.
OWIA | sochi2014.olympics.com.au