MEXICO CITY — Mexico’s projected presidential winner Claudia Sheinbaum will become the first woman president in the country’s 200-year history.

“I will become the first woman president of Mexico,” Ms. Sheinbaum said with a smile, speaking at a downtown hotel shortly after electoral authorities announced a statistical sample showed she held an irreversible lead. “I don’t make it alone. We’ve all made it, with our heroines who gave us our homeland, with our mothers, our daughters and our granddaughters.”

“We have demonstrated that Mexico is a democratic country with peaceful elections,” she said.

The National Electoral Institute’s president said Ms. Sheinbaum had between 58.3 percent and 60.7 percent of the vote, according to a statistical sample. Opposition candidate Xóchitl Gálvez had between 26.6 percent and 28.6 percent of the vote and Jorge Álvarez Máynez had between 9.9 percent and 10.8 percent of the vote. Ms. Sheinbaum’s Morena party was also projected to hold majorities in both chambers of Congress.

The climate scientist and former Mexico City mayor said that her two competitors had called her and conceded her victory.

The official preliminary count put Ms. Sheinbaum 28 points ahead of Mr. Gálvez with nearly 50 percent of polling places reporting.

The fact that the two leading candidates were women had left little doubt that Mexico would make history Sunday. Ms. Sheinbaum will also be the first person from a Jewish background to lead the overwhelmingly Catholic country.

She will start her six-year term October 1. Mexico’s constitution does not allow reelection.

The leftist has said she believes the government has a strong role to play in addressing economic inequality and providing a sturdy social safety net, much like her political mentor, President Andrés Manuel López Obrador.

Ms. Sheinbaum’s victory suggests that the political movement Mr. López Obrador created will live on after his presidency.

His anointed successor, the 61-year-old Ms. Sheinbaum led the campaign wire-to-wire despite a spirited challenge from Mr. Gálvez. This was the first time in Mexico that the two main opponents were women.

“Of course, I congratulate Claudia Sheinbaum with all my respect who ended up the winner by a wide margin,” Mr. López Obrador said shortly after the electoral authorities’ announcement. “She is going to be Mexico’s first (woman) president in 200 years.”

If the margin holds it would approach his landslide victory in 2018. Mr. López Obrador won the presidency after two unsuccessful tries with 53.2 percent of the votes, in a three-way race where National Action took 22.3 percent and the Institutional Revolutionary Party took 16.5 percent.