The 21-year-old was one of just two Brits invited to compete in Aspen with Aimee Fuller finishing eighth in the women's snowboard slopestyle final.
Woods couldn't have asked for a better start on debut as he produced the best score of the qualifying round with an effort of 92.66 in Colorado.
He then placed second with an effort of 91.33 after the first run of the final with American Nick Goepper leading the way after scoring 92.33.
Woods fell on run two while Goepper upgraded his score to 94.00 on the third run while Henrik Harlaut moved into the second after being awarded 92.66.
The Brit, who leads the slopestyle World Cup standings with two victories from two this season, was the last to go in the third run in Aspen.
And, despite posting a best effort on his third run, Woods couldn't overthrow Goepper or Harlaut as he settled for bronze with a score 92.00.
Had Woods won the slopestyle, he would have been the first ever rookie to do so however he overcome a poor practice hours beforehand to take bronze.
"That was the worst practice ever. I didn't land a run, I didn't make a run down the course the whole practice," said Woods after his first run.
"So to come down and put in a 91.33 is making me feel pretty good. I want to go a lot bigger with better grabs and show off why I'm here."