Winnipeg's Braden Calvert scored singles in both the ninth and 10th ends to squeak out the quarter-final victory.
The Swiss had deuces in the third, sixth and eighth ends before the Canadians rallied to win the game.
Canada will now wait for the result of Saturday's semifinal between Sweden and Switzerland to see who it will play in the final on Sunday.
On the women's side, the Canadians remained undefeated going into their page playoff with a 7-1 win over Korea.
Edmonton's Kelsey Rocque opened the game up by scoring three in the seventh end before the Koreans answered with a single in the eighth. The two countries ended the game after the end.
Canada will now play Scotland on Saturday afternoon in its page playoff game with the semifinals going later in the day.