Tracy Fleury was the lone Canadian winner, defeating Russia's Alina Kovaleva 7-5 in women's action.
Olympic champion John Morris, paired with Kalynn Park for mixed doubles, allowed singles in each of the final three ends to lose 6-5 to the Swiss duo of Jenny Perret and Martin Rios.
That collapse came on the heels of Jason Gunnlaugson's own disappointment, who allowed four points in the sixth end and a single in the seventh in falling 8-6 to Bruce Mouat's Scottish side.
Canada swept the opening leg of the World Cup in Suzhou, China, where Olympians Rachel Homan and Kevin Koe won the men's and women's tournaments, while Laura Walker and Kirk Muyres took home the mixed doubles crown.
Action in Omaha continues Thursday, as Morris and Park play Sweden and Korea, Fleury takes on Korea and Gunnlaugson squares off against China and the U.S.