The 33-year-old from Aylmer, Que., beat Amelie Kober of Germany by 3.2 seconds in the final.
"It was my goal to be the first Canadian woman in alpine snowboard to win a World Cup," Calve said. "I haven't been the most consistent person riding out there, but I always felt I could do this.
"To do it finally is really good."
Austrian Marion Kreiner beat Julie Zogg to take third.
Calve was fortunate to make it through the quarter-finals, making up a deficit of 1.22 seconds thanks to a mistake by rival Claudia Riegler of New Zealand to lead by just 0.01 seconds.
"I wasn't thinking about podiums or finishing any position today," Calve said. "I was concentrating on each run. I was trying to do my best all day long.
"It was a tough course. It wasn't the kind of course where people (were) having perfect runs. It was always having mistakes and having to recover quickly from them."
Bob Joncas, the high performance director of Canada Snowboard, said that national team officials were very pleased with Calve's result.
"We are all very excited for Caro," said Joncas in a statement. "It was just a matter of time. In fact, we talked about this last week in Telluride. The coaches were confident that Caro would be on the podium soon and it happened today.
"I am really happy to see Caro reach this milestone both in her career and for Canada."
Roland Fischnaller of Italy won the men's race after Benjamin Karl crashed during his second run.
Karl was 0.09 seconds ahead of Fischnaller after the first run, but the World Cup leader crashed halfway down to hand Fischnaller his first win in the discipline after finishing second twice.
It was Fischnaller's fourth career World Cup victory overall and second this season.
Rok Flander of Slovenia was third after beating Austrian Andreas Prommegger in the small final.
Karl still tops the overall standings, 10 points ahead of Fischnaller.
Jasey-Jay Anderson of Mont-Tremblant, Que., was the top Canadian male, finishing 10th.
— With files from The Associated Press.Suggest a correction