With the players on both sides exhausted, penalties looming and the referee looking at her watch, Morgan looped home a header to put the U.S. in front for the first time after Canada had led three times in a pulsating game at Old Trafford.
The winner came with 122 minutes and 25 seconds on the clock. Sixty seconds later, the match was over.
"I'm still in shock thinking of what just happened," Morgan told reporters. "I don't care who scores as long as we score and that's all that matters.
"I didn't even see it go in. I just try and be in the right position.
"It was a crazy battle. Coming back and back and back. After we scored they scored, it was amazing."
The U.S, who have now reached all five women's Olympic finals, taking gold in 1996, 2004 and 2008, will meet Japan in Thursday's gold medal decider at Wembley Stadium seeking their third successive gold and their fourth in total.
Japan, who overcame France 2-1 earlier on Monday, beat the U.S. on penalties in last year's World Cup final.
Canada's stunned players were in tears at the end and Christine Sinclair, whose hat-trick looked as though it would prove enough for victory, was especially distraught.
But every time she scored, the U.S showed the mark of champions and responded for a fifth successive win in the tournament to claim a place in the gold medal match.
"I'm very proud of this team for making the final," said American Abby Wambach. "I never had a doubt.
"It's exactly what we hoped for [playing Japan in the final] after that World Cup loss."
DELIGHTFUL SLALOM
Sinclair had opened the scoring with a delightful goal after 22 minutes when she slalomed past two defenders after being set up by Melissa Tancredi.
The U.S equalised with a rare goal scored direct from an inswinging corner by Megan Rapinoe that beat goalkeeper Erin McLeod and two defenders at the near post.
Sinclair put Canada back in front after 67 minutes with a firm header to start a wild scoring spree of four goals in 12 minutes - two from each side.
Rapinhoe equalised again three minutes later with a fierce angled drive from the edge of the box that went in off the far post before Sinclair got her hat-trick to put Canada 3-2 up with another header after 73 minutes, her 143rd goal for her country.
But in the 80th minute the U.S equalised for the third time when Wambach scored her 143rd international goal with a penalty after Marie-Eve Nault was harshly judged to have handled in the area by Norwegian referee Christiana Pedersen when a free-kick struck her elbow as she was turning away from the ball.
Extra time looked set to end without a goal after Wambach's header hit the bar in the 119th minute but that was not the last of the drama and deep into added time Morgan headed in a right-wing cross for the dramatic winner.
"The heart we showed today was absolutely huge," said Rapinoe. "It's a little bit of a trademark of our team
