Sinclair scored her second goal of the tournament and 153rd of her career in the opening minute of the second half, after Josee Belanger drew the Korean defence out of position.
Canada improved to 2-0 on the 12-team tournament after beating Scotland 2-0 in Wednesday's opener.
"It was just a great experience for the players," said Canadian coach John Herdman. "The Korean team certainly had a game plan to disrupt the flow and to not allow us to get into rhythm.
"There was a lot of technical fouls, a lot of diving, but that's just the stuff that goes on when you play in big matches and teams are looking for ways to try and beat you."
Canada was reduced to 10 players after 16-year-old Jessie Fleming was sent off in the 66th minute when she picked up her second yellow card.
"It's all experience for her. She's devastated and it's great seeing the emotion," Herdman said. "Jessie was unfortunately in the wrong place at the wrong time, but she learned from that.
'I think that's a situation where a young player gets experience with referees (that are in a certain mindset), and you have to be just that little bit careful."
Sinclair's goal leaves her six away from passing Mia Hamm and into second place for all-time international goals.
The victory stretched Canada's undefeated streak to eight games, and marked the second consecutive shutout for 'keeper Erin McLeod.
"Korea Republic's tactic was to press us as much as possible and kind of be chaotic," McLeod said. "So for us it was a real challenge to find our platforms, to build different ways out of the back and to find our rhythm.
"I am very proud of the resilience that we showed, especially when we went down to ten men, we never let up, and I think that's why we got the result. I am proud of the whole team and that clean sheet is for everyone."
The Canadians wrap up Group A play Monday when they face Italy.
Canada has won the Cyprus Cup three times, and will be gunning for its first title since 2011. They finished second in Group A in last year's tournament and were relegated to the fifth-place match where they beat Ireland 2-1.
It's Canada's final tournament before this summer's FIFA Women's World Cup.