Richardson clocked 10.65 seconds after catching Jamaica's Shericka Jackson (29), who took silver in 10.72.
Jamaican Shelly-Ann Fraser-Pryce, who was seeking a remarkable sixth world 100m title at the age of 36 after an injury-hit season, had to settle for bronze in 10.77.
Richardson missed the Tokyo Olympics in 2021 after testing positive for cannabis then failed to qualify for last year's worlds on home soil in Eugene.
Hugely popular on social media, and always an outspoken and colourfully attired competitor, she was in good form this year, despite Jackson having the fastest time of 10.65 coming into Budapest.
However, Richardson's global championship struggles looked to be continuing when she was left in the blocks in the semi-finals earlier on Monday and she had to dig desperately deep to finish third and scrape into the final as a fast loser.
That came with the consequence that Richardson was given lane nine - never popular with sprinters - but she had a far better start in the final and maintained her focus away from the traffic.
Jackson, in lane four, had opened a clear gap and was still ahead at 80 metres before Richardson swept through and raised her arm in triumph. The American's time matched Jackson's 2023 best and was a championship record.
Richardson is the first US winner of the women's 100 since the triumph in 2017 of Tori Bowie, who died this year aged 32.
MASSIVE TALENT
Coming after Noah Lyles's victory in the men's race on Sunday, the U.S. is firmly back on top of the sprinting mountain after playing second-fiddle to Jamaica for a long spell.
"This is incredibly important because so much has been made of Richardson," said American former 200 and 400 metres champion Michael Johnson.
"She's a massive talent. She didn't make the team in Doha and then came back in 2021 and won the trials. Everyone knew who she was after the ban.
"She's very authentic, she doesn't make excuses. Some people love to love it and some people love to hate it. I think it's great for the sport because she has a personality that is unmatched."
Fraser-Pryce was phlegmatic about her bronze. "Given the circumstances of how I started the season - not bad," she said.
"Being a champion is not all about winning. I'm grateful to have another medal to add to the tally. I won't be running the 200, I'll rest for the 4x100 relay."
It was another frustrating night for Marie-Josee Ta Lou (34) of the Ivory Coast, who has been in superb form in the Diamond League this season but had to settle for another fourth place in 10.81 having twice finished fourth in Olympic 100m finals.
