USA defeated Brazil 1-0 in Saturday's final at the Parc des Princes through Mallory Swanson's goal just before the hour mark.
Hayes only took over as USWNT head coach in May, after leaving her role at Chelsea, and tasted success in her first tournament in charge.
Speaking to Discovery+ at full-time as the celebrations started, Hayes said: "I'm very emotional. It's been a dream of mine.
"I have to thank my dad - he's the one who pushed me to this position, to be able to come and coach an unbelievable group of players."
Hayes' father, who she has long described as a cornerstone in her football career, passed away last year.
"I've got my dad's necklace on," Hayes said post-match. "It's got an American eagle, and he was with me today and that helped.
"The crowd were tremendous. Brazil were tremendous. I'm just speechless. I'm in shock."
The United States have now won five Olympic golds in the women's final - four more than any other nation - with this their first since 2012.
Swanson calmly converted to give USA the victory in the French capital on her landmark 100th senior appearance for her national side.
This latest triumph comes 10 games into Hayes' tenure and puts the poor showings at the 2020 Games and 2023 World Cup in the past.
"They have received me so well and taken on board everything I've asked," Hayes added. "They're tremendous people, players and role models and I love them."
Brazil have still yet to win an Olympic gold in the tournament, with this their third runners-up finish after also falling just short in 2004 and 2008.
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.
The USWNT come into the competition on the back of their earliest exit from a World Cup, losing to Sweden on penalties in the last 16 last year.
The United States arrive in Paris aiming to claim a fifth gold medal in the tournament, last triumphing in the competition at London 2012.
"This team is past that. I think this team is firmly focused on, you know, creating a new history together," said head coach Hayes, reflecting on the 2023 World Cup.
"Our motivation isn't always about righting the wrongs. Far from it. We're excited. We're prepared."
Since Hayes' arrival at the helm in May, she has won three of her four games in charge, securing four clean sheets in the process.
But a goalless draw with Costa Rica in their final pre-tournament fixture has supporters worried about their gold medal prospects.
"The team is exactly where it needs to be at this stage," Hayes assured. "And for us it's just so, so important we continue to focus on that process."
Hayes' side open their Group B campaign against Zambia, who have Barbra Banda in their ranks, the striker who has 12 goals in 12 games for Orlando Pride.
The USWNT are among the favourites to scoop the top award in Paris, with reigning world champions Spain also tipped for gold on their tournament debut.
But Hayes, who won seven WSL titles with Chelsea, feels the age-old assumption of American dominance in the women's game is disrespectful.
Hayes acknowledged that ambition is important, but emphasised that expectations for these Olympics must be amended due to the global growth of women's football.
"I think one of the biggest problems is that too often is that we talk about what we're going to do in Paris or what happens when we get to the semis and finals - I think it's disrespectful to the rest of the world to talk like that," Hayes said.
"I think the game has moved to a point where that isn't the case. It isn't a shoo-in to get somewhere. It has to be earned and there's no given right.
"There's top footballing nations in this tournament. I don't think shocks in the women's game exist any more.
"I think we have to reframe our focus a little bit and have respect for the rest of the world."
Trinity Rodman's 17th minute strike was followed up by a quickfire double from Mallory Swanson to ensure a winning start to Group B.
Swanson's brace came within 66 seconds of each other, netting the fastest two goals by a single player in a major tournament in the team's history.
Lindsey Horan notched two assists in Nice, becoming the fifth player in USWNT history with multiple assists in an Olympic match, joining Mia Hamm, Heather O’Reilly, Megan Rapinoe and Julie Ertz.
Plenty was made of the United States' form heading into the Games, having endured a goalless draw in their final pre-tournament fixture with Costa Rica.
However, their ruthless first-half showing saw Hayes become the first coach to win their first major tournament match in charge of the USA by a margin of three or more goals.
The former Chelsea head coach has also helped her side keep five clean sheets in her first five games in charge, a record only bettered by Greg Ryan (nine).
In spite of their strong showing, Hayes' side underperformed their expected goals (xG) by 1.96, with only eight of their 27 shots on target in the contest.
They missed a total of nine big chances, while also having 58 touches in the opposition's box compared to Zambia's 11.
But having seen other tournament favourites Spain and Germany kickstart their respective campaigns with victories, Hayes is confident her side can improve to claim a record-extending fifth gold medal.
“The first part of the first half was exceptional,” Hayes said. “To come out the way we did, the intention, the intensity, the decision-making, the execution, it should have been at least 5-0 at half-time.
"There was things within our structure I didn’t like. But that’s easy fixes.
“If we had asked me at the beginning of the game ‘would you be happy with a 3-0 winner?’ I probably would have said no.
"However, for the performance and the chances created and finding the back of the net, I don’t want to be too hard on the players.”
The United States continue their Group B campaign on Sunday where they will face Germany, before travelling to Marseille to face Australia next week.
Morgan, who has scored 123 goals in 224 appearances for her country, has appeared at every Olympics for Team USA since Beijing 2008 and won gold at London 2012.
The two-time World Cup winner has not scored in her past eight games for San Diego Wave after recently recovering from an ankle injury.
"Today, I'm disappointed about not having the opportunity to represent my country on the Olympic stage," posted Morgan on social media. "This will always be a tournament that is close to my heart and I take immense pride an time I put on the crest.
"In less than a month I look forward to supporting this team and cheering them on alongside the rest of our country."
Former Chelsea boss Emma Hayes will take charge of her first tournament as the USA manager, and has opted for Trinity Rodman, Jaedyn Shaw, Sophia Smith and Mallory Swanson as her main forward options.
"It was a tough decision, especially considering Alex's history and record with this team, but I felt I wanted to go in another direction and selected other players," Hayes said.
Women's football in Paris begins on 25 July, with the gold-medal match taking place on August 10.
The United States, who won bronze at Tokyo 2020, are drawn in Group B where they will face Zambia, Germany and Australia.
Mallory Swanson was denied a winner in normal time, and though Germany pushed for a late equaliser, Emma Hayes will take charge of a final in her first tournament as the USA's head coach.
Unlike in the group-stage meeting between these sides, which finished 4-1 to the dominant Stars and Stripes, it was Germany who almost took a first-half lead, with Alyssa Naeher reacting quickly to get down and push Jule Brand's low shot wide.
In the 86th minute, Mallory Swanson thought she had snatched a late winner – she latched onto Trinity Rodman's throughball and drilled her low shot across goal into the bottom corner, but the offside flag denied her.
Swanson turned provider five minutes into extra time, as Smith timed her run to perfection, darting in behind to beat Berger for power with a wonderful effort.
There was late drama though, as Naeher brilliantly smothered Laura Freigang's close-range header before Smith was denied a second goal at the other end by Berger's quick thinking.
Data Debrief: Smith's touch of gold
Smith has been in top form during the Olympics and netted her third goal in the competition as she carried her team into their 11th major tournament final.
She is also the second USA player to score in at least three consecutive appearances against Germany, after Abby Wambach - five straight from 2009-12.
Meanwhile, it was familiar heartache for Germany as they were eliminated in the knockout rounds of a major tournament (Olympics and World Cup) for the 10th time, with half of those coming at the USA's hands.
There had been pressure on the USA going into the game after two poor warm-up displays, including a 0-0 draw to Costa Rica, but Emma Hayes' team ran out comfortable winners in the former Chelsea boss' first competitive match in charge.
All of the damage was done in the first half, with the Americans three goals up after just 25 minutes in Nice - Trinity Rodman opening the scoring before a quickfire double from Mallory Swanson.
The four-time Olympic gold medallist started as they meant to go on, hitting the crossbar early on through Rodman and Swanson before the former opened her account for the tournament with a drilled finish past Zambia goalkeeper Ng'ambo Musole.
They doubled their advantage thanks to Chicago Red Stars forward Swanson, and she was on hand again to virtually end the contest just a minute later after good work down the left from Sophia Smith.
Zambia's woes were compounded by a red card dished out to Pauline Zulu, which was initially given as a yellow but was upgraded following a VAR review.
Zulu was adjudged to have been the last player when she brought down 18-year-old Smith, and her dismissal all but ended Zambia's slender comeback hopes.
Data Debrief: History made in Hayes' competitive debut
Though the USA had gone into the contest lacking rhythm and form, which raised hopes of a Zambian upset, it was not to be found, with Hayes' side enjoying complete domination from the start. The USA created chances worth an xG of 4.96 to Zambia's meagre 0.37, while they also had 27 shots to Zambia's eight, with 25 of those coming from inside the box.
Hayes became the first coach to win their first major tournament match in charge of the USA by a margin of three or more goals. Meanwhile, with assists in the 17th and 24th minute, Horan is the first player in USWMT history to record two assists in the opening 30 minutes of a major tournament.
Swanson's goals came 66 seconds apart, making them the fastest two goals by a single player in a major tournament in the team's history.
The United States face Brazil in Saturday's showpiece match at the Parc des Princes in a repeat of the 2004 and 2008 finals, fixtures the US won on both occasions.
Hayes, who was appointed as Vlatko Andonovski's successor in May, has won eight of her nine games in charge (D1), scoring 19 goals and conceding just twice.
The former Chelsea head coach won her seventh WSL title in her final season with the Blues, saying her rapid transition to the international stage has been a "whirlwind".
"Going from my beloved Chelsea to the US women’s national team - what a whirlwind and what a dream all at once," Hayes said.
"To finish my time with Chelsea with another Women's Super League title and then be here competing for gold, honestly this is the stuff dreams are made of.
"I have visualised this moment so many times in my life."
Brazil are seeking their first gold in women's football at the Games, and are boosted by the return of Marta, who will play her final international match on Saturday.
The 38-year-old was shown a red card in Brazil's final group game against Spain for a high foot, resulting in her missing the quarter and semi-finals.
Marta has scored 121 goals in 203 appearances for her nation, becoming the first player to score in five straight Olympics with her brace against Japan.
“When I was a young coach assisting Vic Akers at Arsenal, we played a Women’s Cup two-legged final against Umea, the Swedish champions, and they had a young Marta,” Hayes recalled.
“I remember being on the touchline thinking, ‘Oh my goodness, who is this extraordinary player?’ To witness what she’s done on the pitch, I think she’s been more than a role model and icon, she’s been the best player.
“What she’s done for Brazil or the generation coming through, she’s been an inspiration.”
Team USA have the chance to win a first gold medal since London 2012 after winning bronze in Tokyo three years ago.
Despite winning 16 major titles during her time at Chelsea, Hayes is hungry to add to her trophy cabinet while continuing her unbeaten start in the US dugout.
"I like winning, it's part of my DNA," Hayes said. "I'm with a team that likes winning so of course our goal is to win the gold medal."
The 23-year-old's 95th-minute strike proved the decisive goal of the USWNT’s 1-0 semi-final victory against an injury-stricken Germany.
The result means new coach Emma Hayes will have the chance to go for gold in her first major tournament at the helm.
Smith dedicated her goal to the effort of the whole US squad under the ex-Chelsea boss.
"I just saw an open net in front of me and knew I had to step up in the big moment," Smith told NBC. "I felt I had to do that for this team, we've been working so hard.”
