American Nina O'Brien opened up a 0.13-second advantage after seeing off Kristin Lysdahl, but River Radamus was edged by 0.04sec by Alexander Steen Olsen in the second run in warm, sunny conditions on the Roc de Fer piste.
The third tie saw Paula Moltzan and Thea Louise Stjernesund finish in a dead heat.
That set up a dramatic fourth and final ski-off between Tommy Ford and Norwegian rival Timon Haugen.
Disaster struck, however, when Haugen pre-empted the ramp in the start hut, pushing too early with all his force against the barrier and almost falling onto the slope when the ramp eventually descended.
There was no such mistake for Ford, who was already two gates clear by the time Haugen had righted himself to come through the finish line 1.50sec ahead to ensure gold for the American quartet.
"It was really special," said Radamus, who was part of the US team that finished fourth in the event at the 2022 Beijing Olympics. "I love these team events because it's something we don't get to do all the time; compete as a team.
"It can seem like ski racing is an individual sport but really these people, these guys and girls I am on a team with, they're my family. We are competing and training together all year long.
"So, to get to do it with Tommy and with Nina, Paula, Katie (Hensien) and Luke (Winters) is really special to me. It brings out a different level in me whenever I get to compete like that."
Canada drew 2-2 with Austria in the third-placed ski-off but claimed bronze on time difference.
Valerie Grener, Jeffrey Read, Britt Richardson and Erik Read took bronze by 0.67sec on the back of the first of the Read brothers' crushing win over Dominik Raschner.
The men's and women's parallel races take place on Wednesday.
