Stina Blackstenius' first-half opener had the Scandinavians ahead but Jessie Fleming restored parity with a spot-kick after half-time when VAR intervened for a foul on Christine Sinclair.
Kadeisha Buchanan proved Canada's last-ditch hero with a clearance off the line to take the game to extra-time before Labbe's two saves secured a 3-2 triumph in the shoot-out.
Kosovare Asllani did the initial work for Sweden's opener, dispossessing Canada midfielder Quinn before rolling to Blackstenius, whose first-time attempt benefited from a slight deflection to find the back of the net.
Sinclair, whose 187 goals for Canada make her the all-time leading goalscorer in international football, nipped in front of Amanda Ilestedt and, after consulting VAR, Fleming levelled things up with a low spot-kick.
Asllani spurned a glorious opportunity to seal gold with a minute to go as she saw her effort cleared off the line by Buchanan, sending the final to extra-time and subsequently penalties with neither side finding a winner in the additional 30 minutes.
Sweden failed with their opening penalty, Asllani hammering into the right-hand post, before Fleming coolly slotted in Canada's first to gain an early advantage.
That lead quickly faded away, however, when Ashley Lawrence saw her effort saved and both Nathalie Bjorn and Olivia Schough converted for the Swedes.
Swedish veteran Caroline Seger had the chance to secure glory with the decisive strike, despite Labbe's save against Anna Anvegard, but she blasted over, sending the shoot-out to sudden death after Deanne Rose found the top right corner.
Labbe produced another stop from Jonna Andersson, teeing up Julia Gross for victory and she made no mistake, finding the bottom left corner to seal gold after consecutive bronzes in 2012 and 2016.
In the men's football, Mexico recovered from semi-final shoot-out heartbreak against Brazil to beat Japan 3-1 in Friday's bronze-medal match.
Sebastian Cordova netted the opener and provided the second for Johan Vasquez before Alexis Vega put the game out of the host nation's reach, despite Kaoru Mitoma grabbing a late consolation.
Peter Gerhardsson's side, seeking to go one better than Rio 2016 when losing to Germany in the final, will face the United States' conquerors Canada in Friday's gold medal match.
USA, who had won 36 games in a row against Canada prior to their surprise 1-0 loss earlier on Monday, will now take on Australia for the bronze medal.
Rolfo was on target twice when Sweden ran out 4-2 winners against Australia in the group stage and went closest to scoring in the first half of this semi-final showdown.
The Barcelona star thumped a shot against the crossbar from the edge of the box, though she did not have to wait much longer to find a breakthrough.
A heavily deflected shot from range bounced awkwardly off the wet surface and a backpedalling Teagan Micah parried it onto the crossbar, with the ball looping back into play.
Stina Blackstenius helped the loose ball to Rolfo and the forward hooked a first-time shot away from Micah from four yards to score what proved to be the match-winning goal.
Australia enjoyed spells on top but a 96th-minute red card shown to Ellie Carpenter for dragging down Lina Hurtig signalled the end of the Matildas' hopes of reaching a first final.
Strikes either side of the half-time whistle from Stina Blackstenius and a goal by Lina Hurtig proved the difference as Sweden claimed a famous triumph over their heavyweight rivals, who they defeated on penalties in the 2016 quarter-finals.
The world champions enjoyed a two-year spell without losing before the Group G opener but will quickly have to bounce back if they are to follow up their France 2019 success with Olympic gold.
Rapinoe found it hard to defend the USA's performance as she credited Sweden's hard-working display.
"We did not play a very good game and that is to take nothing away from Sweden, they played a great game," the attacking midfielder said after the match.
"This is the highest level and these are the best teams in the world. Sweden are one of the best ever in Europe and the world. If we don't play well we don’t win these games.
"We want to be a lot better, we played a bit tight and hurt ourselves a lot. You can't say one thing specifically.
"I can't remember the last time we gave up a goal, so to give up three is not great but we know what we need to do to win these games, get out of the group and go from there."
Head coach Vlatko Andonovski refused to panic and is keen to see a response against New Zealand on Saturday.
"It has put us in a big hole and we are the only ones who can get ourselves out of it," he said. "It is not going to be easy. We have to get good results in the next two games, but I know this team will not give up.
"I don’t remember this team losing 3-0 in recent history so it is a bit of a shock, but everyone is positive. We still have games ahead of us. We have got to bounce back. We have to forget this game and focus on the next one."
In contrast, Blackstenius, whose brace sealed a remarkable victory, spoke glowingly of her team, who were without Chelsea captain Magdalena Eriksson for the win.
"I'm very happy about the goals - of course I'm happy to score. The team helped me very much and I think I could have scored at least two more but I don't want to focus on that. I'm just happy and proud of the team", the forward said.
“We developed our game and our play. It's very good and we have so much quality in every player. Every player is also very different from the other. We have so many players that can do good stuff and as a team we are very good. Every player is very happy about going very attacking."
Hedvig Lindahl, though, maintained that Sweden's win was simply a starting point and that Peter Gerhardsson's side had not achieved what they had come to the Games for yet.
"It's just the first group game, we haven't won anything yet. We need to keep going," the veteran goalkeeper said. "If you have one high into lows in the group then this means nothing. But to win against the USA, it's something we can take some confidence from.
"You can go very far in a tournament even if you lose to the USA or whoever you play in the first game, so I don't know how much it means, but we showed the world and ourselves that we can play good against a team like the US and any team."
