Skiing in challenging conditions, Cox was a mere 0.06 points behind the winner while Graham fell short by just 0.95 points.
The women's event was won by Japan's Arisa Murato, with compatriot Miki Ito occupying the other podium position.
The men's moguls competition was won by Russian Alexander Smyshyaev ahead of teammate Denis Dolgodvorov in third.
The women's event was a close fought affair with the three top finishers scoring 18.54, 18.48 and 17.68 points, in a three way battle for victory.
Although Cox wanted the win, she is particularly pleased with the way she skied.
The Australian Institute of Sport/NSW Institute of Sport athlete was the benchmark with her turns, with a point advantage over Murato, and went close to earning the judges' top marks with jumps.
However, the difference between winning and coming second was Murato's speed down the mountain, which has become the hallmark of the Japanese team.
Snow speed has been a focal point for Cox's on-going development and she has been working hard with AIS mogul skiing program head coach Steve Desovich.
"Considering the conditions, there was an improvement but hopefully I can go a bit bigger again and faster in the second competition," Cox said.
"I have a few things I want to work on but overall I was pretty pleased with the way I skied."
Fellow AIS/NSWIS skier Graham was close to Smyshyaev with speed, the Australian matched the Russian with jumps points but was unable to match the winner with turns.
NSWIS athletes David Graham and Sam Hall were also impressive, finishing in fifth and sixth places, a strong result given the quality of the 49 men competing from Australia and overseas.