Lawes and Morris become the first Canadian curlers ever to capture two gold medals in curling - and just the fifth and sixth curlers ever. Boutin actually finished fourth but was bumped up to third after South Korea's Minjeong Choi was disqualified.
Stina Nilsson, of Sweden, won the gold medal. Up next will be a showdown Thursday with the arch-rival Americans, who dispatched the Russian entry 5-0.
Morris earned his second gold, as well, his first coming in the men's event back in 2010 in Vancouver.
"It feels fantastic", said Morris.
The gold-medal victory caps a remarkable run for Lawes and Morris, who only formed a mixed doubles team in December and practised just once together prior to winning the Canadian mixed doubles trials in Portage last month.
Canada came a long way since the announcement three years ago that mixed doubles curling would be an Olympic sport.
Canada broke open a 2-2 game in the third end, capitalizing on a flash by Switzerland's Martin Rios to score a four-ender and seize control.
Switzerland came back with two points of their own in the second to tie things up. They are the first two to win two gold medals.
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The shotmaking abilities of Morris, 39, and Lawes, 29, overcame their lack of experience together as a team.
Formed by husband and wife Alexander Krushelnitskiy and Anastasia Bryzgalova, the Olympic Athlete of Russian Federation rink took the bronze earlier in the day.
But a mistake by the German pushed the Canadian on to the podium.
Norway's Johannes Hoesflot Klaebo produced a stunning final sprint of 3:05.75 to take the men's gold.
"I wanted to make sure it was tight before I threw it", she told CBC Sports during a break in play. "Especially to come that gut-wrenching feeling of being behind (Eitberger) and probably in a fourth spot again and to have that flip around on me and be suddenly in a medal spot is so unbelievable".
Canada edged ahead with an early four-pointer before steadily growing their lead through some tactical stones, although a nervous performance from Switzerland's Rios helped their cause.
The awarding of the bronze medal to Boutin, of Sherbrooke, Que., capped what began as a disappointing day for the Canadian team. "The power play is a chance to get multiple points and it really favours the team with the hammer". She made contact with Yara Van Kerkhof of the Netherlands while jockeying for position. "I don't think even the first one [her Sochi 2014 gold] has sunk in and I don't think it ever will". "But short track is short track, so sometimes you have to deal with what is happening".