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.
Hayes, who took charge of the USWNT in May, has won six of her seven games at the helm, conceding just twice in that time.
The USA, along with Spain, are the only two sides to win all of their games at the Paris Olympics so far and will have great confidence heading into the knockout stages.
Trinity Rodman's second goal of the tournament two minutes from half-time sent them on their way, as Korbin Albert's 77th-minute strike saw her become the the youngest substitute scorer for the USA at the Olympics since Mallory Swanson against Colombia in 2016.
Alanna Kennedy's late strike would take some of the shine off an otherwise impressive display, but the United States dominated from the first whistle.
Hayes' side accumulated an expected goals (xG) of 2.82 from the 21 shots they had compared to Australia's 0.46, ending the match with 71.7% possession.
The United States will face Japan in the quarter-finals of the tournament, while Australia face a nervous wait to see if they qualify as one of the best third-placed teams.
In the other Group B encounter, Germany confirmed their place in the knockout stages alongside the United States following a comfortable 4-1 win over Zambia.
Lea Schuller opened the scoring in the 10th minute, with Klara Buhl adding the Germans second of the afternoon shortly after the restart.
Zambia captain Barbra Banda threatened to make the encounter a more difficult affair with her fourth goal of the tournament two minutes later.
But Schuller and Elisa Senß would add further gloss to the performance soon after to secure second place in the group.
Germany's triumph sets up a meeting with the runners-up of Group A, which could be either one of Colombia, France, Canada or New Zealand.
