The Americans came back from a goal down three times to ouster Canada and earn a trip to face Japan for the gold medal.
Sinclair scored the first of three goals in the 22nd minute and gave Canada the 1-0 lead, one that would hold up for the remainder of the initial 45 minutes.
USA fullback Rapinoe got her side back into it after the break when Rapinoe fired in a low hard corner on goal that edged its way over the Canada line to tie the game at 1-1.
The teams then traded three goals later in the second half as Sinclair got her second from a header in the 68th minute. Rapinoe then matched her counterparts pair with a brace of her own from a rocket shot in the 70th minute. A mere three minutes later Sinclair netted another headed goal to complete her historic hat trick.
The goals kept coming as the USA was awarded a controversial penalty kick in the 79th minute, which Abby Wambach calmly converted to tie the score at an amazing 3-3.
Neither side was able to take the lead in the game's final 10 minutes so extra time was needed to determine who would face Japan in the gold medal game. And as it has done so many times before, the USA was pull out some late-match heroics.
All of extra time had wound off the clock before Morgan was able to head home the winner in the 122nd minute. A deep cross from Heather O'Reilly was expertly placed by Morgan who ultimately gave the Americans the 4-3 victory.
The result sends the United States into the Olympic final for the fifth straight time. It will face Japan in a rematch of the 2011 Women's World Cup Final.
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