Two-time runners-up Brazil were seeking their first gold and thought they had taken the lead early on at the Parc des Princes, only for Ludmila's strike to be rightly ruled out for offside.
Swanson used her pace to break clear soon after but was unable to get the better of Lorena, with neither side truly going close to finding a breakthrough in the first half.
The Selecao suffered a blow early in the second period when Vitoria Yaya was forced off the field on a stretcher and their outing soon got worse when Swanson fired home.
Chicago Red Stars forward Swanson held her line well, latched on to Korbin Albert's throughball and calmly slotted past Lorena with 57 minutes on the clock in Paris.
Brazil could have taken a game of few clear-cut chances to extra-time, but an unmarked Adriana saw her header kept out by Alyssa Naeher in stoppage time as Emma Hayes' team claimed gold.
Data Debrief: Swanson the golden girl for USA
Swanson was denied a dramatic goal on her 99th cap for the USA in their 1-0 extra-time win over Mexico in the semis, but she marked her century in the perfect manner as she scored for the fourth time this tournament.
The 26-year-old is the 44th player in USWNT history to reach that milestone and is the youngest since Amy Rodriguez (25) did so in 2012.
The USA's fifth Olympic gold, coming 12 years after their last success in London, takes them four clear of Germany, Canada and Norway as comfortably the most successful side in the competition.
Salma Paralluelo netted two late consolations, but the damage was already done as Irena Paredes' own goal, Gabi Portilho, Adriana and Kerolin denied the world champions a place in the gold-medal match.
Brazil were still without legend Marta after she lost her appeal over a two-game ban, but her team-mates ensured she would bow out of Olympics action with the chance to play for the top prize.
Spain were behind after just six minutes as Cata Coll's poor clearance struck the unfortunate Paredes on the side to send it cannoning over the line.
La Roja struggled to gain a foothold, and Brazil doubled their advantage on the stroke of half-time thanks to Portilho's composure when one-on-one with Coll.
Adriana made sure of their progress off the bench in the 71st minute – she struck the bar but was then on hand to head Portilho's nod-down in from close range.
That sparked Spain into life and Paralluelo's looping header snuck in at the far post before Alexia Putellas rattled the crossbar and forced Lorena into a smart stop to slow down the comeback attempts.
Kerolin delivered the hammer blow in the opening seconds of stoppage time by sliding her low shot through Coll's legs, though Paralluelo would have the last say, poking in her second of the match from close range in the 112th minute.
Data Debrief: World champions denied
Spain were aiming to become the first side to win Olympic gold after lifting the Women's World Cup, but they just fell short at the final hurdle after being outplayed.
La Roja's late flurry of chances meant they finished with more shots than Brazil (26 to their 17) and more on target (11 compared to seven).
Brazil's dominance showed in other ways though, as they created an expected goals (xG) of 3.27 and had a total of six big chances, as they secured a meeting with the United States in the final on Saturday.
