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.
"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."
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
