The Americans had a 2-0 lead and seemed all but certain to end Canada's 19-game Olympic winning streak in women's hockey, but Brianne Jenner gave her team a chance by scoring with 3:26 left in regulation.
With just under two minutes to go and Canadian goaltender Shannon Szabados having left the ice for an extra attacker, the United States almost clinched the game.
A pass to the point was gathered in by a Canadian player who, as she was backing up, collided with one of the referees. An American poked the puck away and put enough force on it to send it all the way down the ice.
Had the puck made it into the empty Canadian net, the United States would have had a two-goal lead. Instead, it struck the left post and bounced away.
Given that reprieve, Poulin tied the game with 55 seconds left on the clock.
When an Olympic medal is on the line, teams play 10 minutes of four-on-four sudden death overtime before going to a shootout.
Poulin, however, ended the overtime before it reached the halfway mark. That gave Canada its fourth consecutive gold medal.
In the third-place game, Jessica Lutz scored the go-ahead goal in the third period to lift Switzerland to the bronze medal with a 4-3 win over Sweden.
The Swiss overcame a 2-0 deficit with four consecutive goals in the closing period to win a medal for the first time since women began playing hockey in the Olympics 16 years ago.
Sweden held a 2-0 advantage going into the final period, having come up with goals from Michelle Lowenhielm and Erica Uden Johansson.
Switzerland tied the game during the first 6:13 of the third period with scores coming from Sara Benz and Phoebe Stanz.
Lutz then put the Swiss in front to stay with 6:17 to play and Alina Muller produced what turned out to be the game-winning goal with 1:07 remaining.
Just 23 seconds later Pernilla Winberg's goal gave Sweden hopes for a late comeback, but Swiss goaltender Florence Schelling held on the rest of the way.
Schelling made 28 saves while Sweden's Valentina Wallner stopped 22-of-26 shots. Sweden had won a medal in each of the last three women's Olympic hockey tournaments.