The jubilant crowd of an estimated quarter-million people had barely cleared out of the victory party in Chicago when President-elect Barack Obama began shaping his administration.
Obama offered the job of chief of staff to Rep. Rahm Emanuel, ABC News' Jake Tapper reported today.
Emanuel, a veteran of President Clinton's administration and a close political ally of Obama's from Chicago, hasn't immediately given his answer.
Obama likes that Emanuel knows policy, knows politics and knows Capitol Hill and has told associates that he knows Emanuel will "have his back," ABC News' chief Washington correspondent George Stephanopoulos said of the offer.
The Illinois senator amassed 338 electoral votes to McCain's 162, although three states -- Missouri, Indiana and North Carolina -- remain too close to project.
In a Rose Garden statement today, President Bush congratulated Obama on his "impressive" victory and noted the historic significance of electing the country's first black president.
"No matter how they cast their votes, all Americans can be proud of the history that was made," Bush said.
"It will be a stirring sight to watch Barack Obama; his wife, Michelle; and their two beautiful daughters step over the threshold of the White House," he said.
The president acknowledged "we are embarking on a period of change in Washington" and promised his "complete cooperation" in the transition in the next two months.
The sweeping triumph, which included winning six states that had voted Republican in 2004, triggered euphoric crowds to turn out in Chicago, where Obama claimed his victory, as well as in New York City's Times Square and Harlem, and on Pennsylvania Avenue in Washington, D.C.
By the time the dancing in the streets had ended, names were being bandied over the airwaves about whom Obama would name to his Cabinet.
The president-elect has signaled that he will rely heavily on former members of Clinton's administration and that he plans to include several Republicans on his team.
His transition team will be headed by Clinton veteran John Podesta, his current Senate chief of staff Pete Rouse, and Obama confidante Valerie Jarrett. Tapper reports that the team will include former Clinton Cabinet members William Daley and Fredrico Pena, as well as Arizona Gov. Janet Napolitano.
"If people think there is a direction, a vision, a plan that we're moving forward, you can change the psychology, help the markets to settle down," Clinton's former chief of staff Mack McLarty told "Good Morning America."
Timothy Geithner, president of New York's Federal Reserve Bank, and Larry Summers, the former treasury secretary in the Clinton administration, are believed to be the leading contenders.
"We're going back to an old role for secretary of treasury where they are the chief economic spokesman and chief economic formulator," Ken Duberstein, President Reagan's former chief of staff, told "GMA."
Stephanopoulos said that Obama is also likely to appoint an emergency economic committee that could include heavyweights like former Treasury Secretary Bob Rubin; former head of Clinton's council of economic advisers Laura Tyson; maybe former Federal Reserve Board chair Paul Volcker; and maybe even billionaire investor Warren Buffett.
Nevertheless, American and European markets fell slightly Monday morning and U.S. markets fell upon opening.
There is pressure on Obama to quickly designate other top spots and he is expected to move fast to designate secretaries of defense and state.
"Campaigns are long and arduous. Transitions are fast and chaotic," McLarty said.
"He needs to put everything in place so that he can hit the ground running on Jan. 20," Duberstein said.
Obama is expected to move quickly to name three or four top spots in his administration: chief of staff and the secretaries of treasury, defense and state.
Stephanopoulos reported that Obama is also committed to having Republicans in his administration and that he will likely reach out to McCain to find some issues they can work on together.
Defense Secretary Bob Gates is expected to be asked to stay on the job.
"It would make good sense for our country and the beginning of an Obama administration," Duberstein said.
Republican Sen. Chuck Hagel may be a backup contender for the job.
Republican Sen. Dick Lugar is believed to be on a short list for secretary of state, but he must contend with the ambitions of Massachusetts Sen. John Kerry and New Mexico Gov. Bill Richardson.
Congratulations poured for Obama from around the world.
The president of Kenya, the home country of Obama's father, declared a public holiday in Obama's honor.
In Indonesia, where Obama lived as a child, hundreds of students at his former elementary school erupted in cheers when he was declared winner and poured into the courtyard where they hugged each other, danced in the rain and chanted "Obama! Obama!"
Some of the congratulations came tinged with warnings.
Russian President Dmitry Medvedev applauded Obama's victory and said he hoped the two countries could have constructive dialogue. Afghanistan's president accompanied his message with a demand that Obama put an end to civilian casualties by American forces.
ABC News' Claire Shipman and Jake Tapper and The Associated Press contributed to this report.