Koe dropped a 9-6 decision to Japan's Yusuke Morozumi in the early draw before improving to 2-1 with a 9-6 victory over China's Rui Liu.
"Obviously it's a long week, but you don't want to get yourself behind the eight-ball early, so obviously it was important to get some stuff figured out," said Canadian third Pat Simmons. "It was a good, gutsy win for us, for sure."
The Calgary-based rink opened the tournament Saturday with a 6-3 win over Denmark.
Norway's Thomas Ulsrud and Germany's John Jahr led at 3-0 after five draws. Canada was tied with Japan, Switzerland and Sweden at 2-1.
China picked up a steal of one in the fourth end when Koe's open takeout attempt overcurled. Canada pulled even at 3-3 with a deuce in the fifth and stole three points in the sixth end when Liu's draw was wide.
Koe added a deuce in the eighth end and one more point in the 10th for the victory.
"That’s big," Koe said. "I mean, you don’t want to get 1-2 this early in the week. It (would) be a long week, for sure."
Teams are still adjusting to the ice conditions in the warm, humid Chinese capital. There have been some uncharacteristic misses over the first weekend of competition, such as Koe's mistake in the fourth end against China.
"I didn't think I threw it too bad in the fourth," said Koe. "Sometimes you throw it and you think you know what it'll do and it still doesn't do that. So that's frustrating. We're used to a little more out of ourselves, so we just have to realize that it's tough for both teams and we have to stick with it."
The Canadian icemaking tandem of Hans Wuthrich and Greg Ewasko have had some challenges — including a contaminated water supply — in their effort to create top-flight conditions.
"You just have to accept that there's going to be some nuances on each sheet," Simmons said. "If you can get control of the game and make the other team play those hard shots, then it's obviously a big advantage.
"We were able to do that and make him shoot against a few of our rocks the one end (the seventh) and we were lucky enough to get the steal. Yeah, we'll take it."
The Czech Republic, Russia, China and the United States were 1-2 while Denmark and Scotland remained winless at 0-3.
The Canadians will play Russia and the Czechs on Monday.
Koe won gold in his only previous appearance at the world championship, taking top spot at the 2010 tournament in Cortina D'Ampezzo, Italy.
Canada has won gold three of the last four world championships. Jeff Stoughton was victorious in 2011 in Regina and Glenn Howard won at Basel, Switzerland in 2012.