On Wednesday, Obama was widely seen as having lost his first debate with Romney.
Jack Welch, the retired former CEO of General Electric, said on Twitter: "Unbelievable jobs numbers ... these Chicago guys will do anything ... can't debate so change numbers."
But the unemployment data is calculated by a government agency, the Bureau of Labor Statistics, under tight security and with no oversight or input from the White House.
Keith Hall, a former commissioner of the BLS who was appointed by President George W. Bush, said the numbers could not have been manipulated.
"It's impossible to do it and get away with it," he said. "These numbers are very trustworthy."
Economists offered reasons not to read too much into them, though. Most of the increase in employed Americans came from those who had to settle for part-time work: 582,000 more people reported that they were working part-time last month but wanted full-time jobs.
That is the biggest increase in so-called underemployed Americans since February 2009, during the depths of the Great Recession.
Economic troubles in Europe and Asia also may be taking a toll on American factories. Manufacturing employment dropped by 16,000 in September after falling by 22,000 in August.
Factory hiring had been a source of economic strength the past two years: Factory jobs rose last year at the fastest pace since 1997.
But Europe's ongoing economic crisis, along with a slowdown in China, means demand for U.S.-made goods is drying up and "the days of robust manufacturing payrolls growth are likely behind us," said Chris Jones, an economist at TD Economics.
The unemployment rate has fallen from a peak of 10 percent in October 2009. But a big part of the drop over the past three years came because so many Americans stopped looking for work, so they weren't counted as unemployed.
Some were retiring baby boomers. Others were so discouraged by the weak job market that they stopped putting out resumes.
Economists were pleased with September's drop in unemployment because it happened for the right reasons: More Americans got jobs. And the work force grew by 418,000, the most since February, suggesting people are more optimistic about finding jobs. Because 873,000 more people did find work, the number of unemployed fell by 456,000. And that decline pushed the unemployment rate down.
Arthur Nazh was not surprised to hear that the unemployment rate fell. Business has improved this year at the kiosk where he sells souvenirs at a mall in Arlington, Va.
The economy "is getting better," said Nazh, 27, who employs six people and said he plans to vote for Obama because the economy needs more time to heal. "People are starting to buy more, spend more money."
Obama and Romney campaigned Friday in Nazh's home state and others that could tip to either candidate and determine the outcome of the election. Romney released three ads, mostly focused on jobs.
"These are tough times in this community," Romney, the former Massachusetts governor, told a rally outside a construction equipment store in Abingdon, Va. "We're going to bring back jobs and bring back America."
At a campaign stop in Cleveland, Obama declared: "We are moving forward again."
"Today's news should give us some encouragement," the president told thousands at Cleveland State University. "It shouldn't be an excuse for the other side to try to talk down the economy just to try to score a few political points."
The political back-and-forth over the unemployment numbers underscored the centrality of jobs to the election after a year in which the economy has been difficult to read.
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