The 23-year-old, who led after the first run, held her nerve in the second run to secure her 18th World Cup podium in front of a noisy American crowd.
Austria's Julia Scheib finished second, 0.96 seconds behind Robinson, narrowly missing out on a second World Cup win. Norway's Thea Louise Stjernesund completed the podium in third place.
"It was definitely a day of two different runs. The first one I felt in control and so smooth, the second felt out of control, like I was recovering the whole time. I was pretty shocked to finish first," Robinson said.
"I was really not feeling that confident coming into this race, so I'm really proud to have followed my instincts today."
American Mikaela Shiffrin, the five-time overall World Cup champion, finished 14th after a poor first run.
The 30-year-old has been rebuilding her form after suffering a puncture wound in her abdomen and severe muscle trauma in a crash during a giant slalom in Killington, Vermont, a year ago.
Shiffrin, who secured her 103rd World Cup win in Austria last week, is a key hope for the US team ahead of the Milano-Cortina Olympics in February.
The Copper Mountain event continues on Sunday with the women's slalom. The circuit then moves to Mont-Tremblant, Canada, next month.