An Alex Morgan header in the 123rd minute was the difference, a goal scored in the final minute of added time.
The Americans move on to face Japan in the gold-medal match on Thursday while the Canadians will play France for bronze. A win would give Canada its first traditional team sport medal since the men's basketball squad earned a silver in 1936.
Canada was looking for its first win against the U.S. since March of 2001, a span of 27 games, but a pair of goals from Megan Rapinoe, and a late penalty kick from Abby Wambach got the powerful American squad to extra time, where Morgan came through with her late-game heroics.
Wambach's penalty came after the Americans were awarded a free kick outside the Canadian box by Norwegian referee Christiana Pedersen when goalkeeper Erin McLeod was whistled for holding the ball for more that six seconds. Marie-Eve Nault was charged with a handball in the penalty area on the ensuing kick.
Canadian coach John Herdman was livid with Pedersen.
"She'll have to sleep in bed tonight after watching the replays, she's got that to live with," he said "We'll move on from this, I wonder if she'll be able to."
McLeod said Norwegian referee Christiana Pedersen told her that she had held the ball for 10 seconds.
"Not even close," McLeod replied.
Sinclair pleaded with Pedersen to reconsider.
"She actually giggled and said nothing," Sinclair said. "Classy."
Sinclair felt the Canadians were robbed of a victory.
"Obviously, we're disappointed and upset. We felt that the referee took it away from us, so, yes, we are disappointed. We feel like we didn't lose, we feel like it was taken from us. It's a shame in a game like that, which is so important, that the ref decided the result before the game started."
In fact, every one of the Canadians was seething.
"I just don't think any of us could believe what happened, that's why we didn't leave [the field] probably," said Melissa Tancredi, her lip quivering as she tried to fight off the tears. "I just didn't want to leave because I just couldn't believe that's what happened. That was our game, that was our win to have. And it was just taken away."
Frantic opening minutes
The Americans held the Canadians in their own end for the a frantic first 15 minutes, but they weathered the storm to finally settle down before Sinclair's goal.
Herdman, a Newcastle native who BBC announcers have been calling "Geordie John," had complained just a day earlier about the U.S. using "highly illegal tactics" with their physical play on set pieces.
"Their coach prepared them very well," Wambach said. "He had a very good tactic [Sunday], by making it a media [event] to say that we do illegal stuff. I give him credit for that because it's something that he was trying to do to rally his team around him."
But on a night of two heavyweights slugging it out, the Canadians doled out as many bruising tackles as they took Monday, keeping the officials on their toes.
Sinclair opened the scoring in the 21st minute off a wonderful strike to beat American goalkeeper Hope Solo. Taking a short pass from Melissa Tancredi inside the six-yard box, Sinclair juked past a defender and beat Solo with a sizzling shot to the lower right corner of the net.
Canada took the 1-0 lead into the second half, but Rapinoe pulled the Americans level in the 54th minute with a corner kick that seemed to fool the Canadians. Her curling kick went through the legs of a Canadian defender and over the line without being touched.
A Sinclair header put the Canadians back on top in the 67th minute, but that lead lasted all of three minutes. Rapinoe replied with her second of the game in the 70th, scorching a strike off the post and past Canadian keeper Erin McLeod.
Americans keep coming back
That tie also lasted a mere three minutes, as Sinclair completed her hat trick — goal No. 143 of her illustrious career — in the 73rd minute with a beautiful header past Solo and her defenders.
"Christine, to come and score a hat trick of a semifinal of an Olympic Games against our biggest rivals and not to come away with something … there something that isn't right about it," Herdman said.
A free kick within the Canadian penalty area led to a handball call on Canada’s Marie-Eve Nault and a subsequent successful penalty kick for Wambach in the 80th minute.
Wambach had a chance to put the game away in the 85th minute but she deflected a hard pass from Alex Morgan just wide of the Canadian goal, setting up extra time.
Wambach also had some of the best chances in extra time, with her header in the 119th minute glancing off the Canadian crossbar, foreshadowing Morgan’s winner just five minutes later.
"Even when they scored their third goal, there was something in me that knew that we had more, that we could give more," Wambach said. "I don't know what that means, quite honestly. I don't know if it's just confidence until the end, but this team has a belief in itself, even when the going gets rough."
The Canadians were eliminated by the U.S. in their Olympic debut four years ago in Beijing, losing 2-1 in the quarter-finals in a game that went to extra time.
Their thrilling ride through the Summer Olympics comes just a year after the team was in total disarray, finishing 16th of 16 teams at the women's World Cup that led to the hiring of Herdman.
The Americans had dominated their opponents in their run to the Olympic semis, going undefeated and allowing just two goals, while Canada went 2-1-1, defeating Great Britain 2-0 to book their spot in the semis.
World Cup champs wait in the wings
Next up for the Americans are the defending World Cup champions from Japan, who defeated France 2-1 at Wembley Stadium in London earlier on Monday.
Yuki Ogimi and Mizuho Sakaguchi each scored for Japan, but France substitute Eugenie le Sommer scored in the 75th minute to put the team within one.
A minute later, Elise Bussaglia could have equalized from the penalty spot following a foul, but she shot wide of the right post.