Koe started the day with a 9-1 blowout victory over Russia's Evgeny Arkhipov at Capital Indoor Stadium and then came back for a 10-6 win over Jiri Snitil of the Czech Republic in the late draw.
"For sure, you have to pull some games out," said Koe. "We weren't struggling as bad (as on Sunday). The sheet was a lot better than yesterday, but no excuses. The Czechs were playing really well and making some big shots.
"I was proud of the guys. We really stayed in there, tried to stay positive and keep it close."
Koe, who suffered his only loss to Japan's Yusuke Morozumi on Sunday morning, improved to 4-1 after eight draws. Norway's Thomas Ulsrud was the lone undefeated team at 5-0.
Germany, Japan, Sweden and Switzerland were tied for third place at 3-2 in the 12-team round-robin standings.
The Calgary-based team of Koe, lead Nolan Thiessen, second Carter Rycroft and third Pat Simmons shot 96 per cent as a team against Russia. Thiessen and Rycroft both scored perfect games.
"That felt way better," said Rycroft. "We threw some stinkers yesterday, and we were lucky to get away with 1-1 (on the day). Finally today we felt like we got a bit of a handle on things.
"We were throwing them properly. Still missing a few, but at least we're throwing them with conviction. The results? You can't worry about them too much because you still get caught on certain spots."
In the late game, Snitil controlled the early ends and built a 6-3 lead. Koe scored a pair in the sixth end and pulled even with a steal in the eighth on a nice hit and roll.
The Canadians followed up with arguably their finest end of the tournament, making eight perfect shots in the ninth end. Koe, sitting four and guaranteed a steal, forced Snitil to attempt a wide outturn draw to bite the outside of the four-foot to limit the damage.
However, the Czech rock caught debris outside of the house and ground to a halt, giving the Canadians a steal of four and ending the game.
"That ninth end was real unfortunate for them, but we were stealing one at worst," Koe said. "We would have been one up coming home and it would have been anybody’s ball game, but this was a big one."
Even as the Czechs built their lead through five ends, Koe said his teammates never lost their focus.
"We've talked with our coaches about it off the ice," he said. "If we're going to lose, let's at least keep battling and stay positive. We did that very well in that game. It's easy to get frustrated after the first few ends, but we hung in there and I'm really proud of the guys for battling that hard."
Canada will play the United States and Norway on Tuesday.
Round-robin play continues through Thursday. The playoffs begin Friday and the medal games are scheduled for Sunday.
Koe won gold in 2010 at his only previous appearance at this event. Canada has won gold in three of the last four years.