Board of Supervisors President London Breed made San Francisco history Wednesday as she was assured of becoming the first African American woman to be elected mayor.

“I’m so hopeful about the future of our city. I’m looking forward to serving as your mayor, and I’m truly humbled and truly honored,” Breed said in a brief appearance on the City Hall steps after the only candidate with a chance of catching her, former state Sen. Mark Leno, conceded the race.

“Whether you voted for me or not,” Breed said, “as mayor I’ll be your mayor, too.”

Breed’s victory sweeps aside a stubborn political barrier for women of color in San Francisco, and it will make the city the biggest in the U.S. to have a female mayor.

It also marks a crowning achievement in a remarkable life story. Raised in poverty by her grandmother, with a sister dead from a drug overdose and a brother in prison, Breed held up her accomplishments as a beacon for anyone struggling to break free from challenging circumstances.

“The message that this sends to the next generation of young people growing up in this city is that no matter where you come from, no matter what you decide to do in life, you can do anything you want to do,” Breed said,

Politically, her election is a victory for the city’s more moderate political faction. Although there were few major policy disagreements among the main candidates, progressives had lined up behind Leno and Supervisor Jane Kim. Each told their voters to list the other as their second pick in the ranked-choice balloting in an effort to elect the city’s first progressive mayor since Art Agnos was voted out of office in 1991.

Breed, 43, will become mayor at a pivotal moment for San Francisco, one in which the city is confronting an epidemic of homelessness, trash-strewn streets and a housing shortage that threatens to squeeze middle- and low-income residents out of the city.

What differences there were between Breed and opponents were most stark when it came to housing. Breed earned the support of pro-development forces — who now include a group of largely young activists who call themselves YIMBYs, for Yes In My Back Yard — with her pledge to “build more housing and build more housing, faster.”

Leno, a 20-year veteran of San Francisco politics who had hoped to be the city’s first openly gay mayor, said he had called Breed to congratulate her and “to wish her every success — personally and professionally — in her new job as mayor of San Francisco.”

He called Breed “a remarkable young woman. ... She is going to do a very fine job and we wish her all the best, because her success is San Francisco’s success.”

Breed thanked Leno and paid tribute to the late Mayor Ed Lee, whom she called a “good man” who had “worked to move the city forward.”

Lee died of a heart attack Dec. 12. Breed will serve out his term, which runs to January 2020. She has not said whether she plans to run for re-election next year.

Leno held a slim lead in the ranked-choice balloting after election night last week, but Breed soon pulled ahead and steadily widened the gap. On Wednesday she had a 2,177-vote advantage over Leno with about 6,690 ballots still to be counted, according to the city Department of Elections.

Leno said he saw no point in waiting for the additional ballots to be tallied, as the trend in Breed’s favor was clear.

Kim, who had said last week after she was eliminated from contention that she looked forward to working with “Mayor Mark Leno,” congratulated Breed on Wednesday.

“I’m proud to live in the largest city in America with a woman as mayor,” Kim said.

As Board of Supervisors president, Breed became acting mayor after Lee’s death. But she lost the job after a month when her board colleagues voted to make Mark Farrell mayor on an interim basis until the special election to pick Lee’s successor.

In a statement Wednesday, Farrell offered his “sincere congratulations to Mayor-elect London Breed on her election victory. I commit my full support, both personally, and through my staff, to make this transition between our administrations as smooth as possible.”

Breed portrayed her life’s trajectory — starting from an impoverished childhood in Western Addition public housing — as an uplifting example to a city that is struggling with issues of income disparity, as well as a sign to others who come from disadvantaged backgrounds that they can aspire to great things.

Breed got her start in politics during the 1999 re-election campaign of Mayor Willie Brown, who persuaded her to take the job of executive director of the African American Art & Culture Complex in 2002. Ten years later, she defeated Lee’s appointed supervisor in District Five and held the job ever since, winning the board presidency in 2015.

In early January, Breed brought 10 young girls with her to the city Department of Elections to watch her fill out the paperwork that would officially launch her bid for San Francisco’s top job.

“I wanted these young ladies to know that if I can be mayor, they can be mayor, too,” Breed said.

Her ascent to the mayor’s office will also lead to a number of changes at the Board of Supervisors in the coming weeks.

One of the board’s first tasks will be to elect a new president to finish out her term, which ends in January. Breed will also appoint her successor for the District Five board seat, which includes the Western Addition, Haight-Ashbury and other central neighborhoods. She is likely to name a replacement in time for that person to vote on a board president.

The board itself will look very different after the November election. Seats in Districts Two, Four, Six, Eight and 10 are up for grabs, and only the District Two and District Eight elections feature an incumbent.

Dominic Fracassa is a San Francisco Chronicle staff writer. Email: dfracassa@sfchronicle.com Twitter: @dominicfracassa