The nation’s jobless rate dropped to its lowest point in nearly four years in September. And unlike some recent declines, this one happened for the right reason: not because people gave up looking for a job, but because far more people reported having one.

It is a surprising improvement in a job market that had appeared listless in recent months. Although employers added a modest 114,000 jobs in September, the unemployment rate dropped sharply, from 8.1 to 7.8 percent, the government reported Friday.

Unemployment is at its lowest level since President Obama took office in January 2009, offering him a political boost just days after his performance was widely judged as lackluster during a debate against GOP rival Mitt Romney.

The government said hourly wages were up and employees worked more hours in September, meaning they were taking home bigger checks. Overall, the ratio of the American population with a job reached its highest level since May 2010.

The drop in the unemployment rate was bolstered by revisions reflecting that employers had added 86,000 more jobs than previously known in July and August, recasting the troubling summer lull in job creation to a season of solid employment gains.

Although the report offered a brightening picture of the nation’s labor market, the overall rate of job creation remains less than robust. In addition, unemployment remains far above normal levels, and many millions who have jobs are not working full time.

“While the September employment report was more encouraging than the ones we have seen in recent months, the job market is still a long way from rosy, good health,” said Gary Burtless, a Brookings Institution economist.

This year, employers have added only slightly more jobs per month than are needed to keep pace with normal labor-force expansion, and slightly fewer than the 153,000 average monthly gain the nation experienced in 2011. Also, the number of Americans working part-time — even though they want full-time jobs — rose sharply last month to 8.6 million.

With economic growth creeping along after showing signs of more vigorous expansion last year, some economists were skeptical of the magnitude of September’s unemployment decline. A few even predicted that the jobless rate would tick up in the coming months.

“This was a pretty good report, but the drop in the unemployment rate was just too good to be true and probably overstates the degree of improvement in the job market,” said Stuart G. Hoffman, chief economist for PNC Financial Services Group.

Even at 7.8 percent, the joblessness rate remains high by any historical standard. And it could be years before the economy returns to full employment.

But Bernard Baumohl, chief global economist for the Economic Outlook Group, said the dichotomy between the recent steep decline in the jobless rate and the slow economic growth in recent months could mean that the economy is poised to take off.

Economic “growth may turn out to be stronger than most economists currently forecast,” he said. “ . . . Employers are, thus, cautiously turning more optimistic about the economy in 2013 and becoming less apprehensive about hiring.”

There is evidence that consumers are feeling better about the state of the economy: The Conference Board’s consumer confidence number was up in September, as was a similar University of Michigan survey of consumer sentiment.

The new jobs report was heartily embraced by Obama, who had left many of his supporters feeling anxious after Wednesday night’s presidential debate.

“The unemployment rate has fallen to the lowest level since I took office. More Americans entered the workforce, more people are getting jobs,” Obama said during a rally at George Mason University.

Romney played down the report, saying it reflected, at best, halting improvement. “This is not what a real recovery looks like,” Romney said in a statement. “We created fewer jobs in September than in August, and fewer jobs in August than in July.”

The September unemployment rate fell dramatically despite the mediocre number of 114,000 jobs created. That’s because the government’s survey of American households — which determines the jobless rate — reported substantially stronger employment gains than a survey of employers — which establishes the overall job number.

Some analysts say this divergence may have been a fluke, while others said it could signal that gains in self-employment and in business start-ups were not captured in the overall job figure.

The two numbers should even out over time, economists said. About 873,000 more Americans reported having jobs in the survey of households, the largest increase since 1983. About 456,000 fewer people reported not having a job but wanting one.

“The rule of thumb when the two surveys tell different stories in the same month is to give much more weight to what” employers say, said Heidi Shierholz, an economist at the Economic Policy Institute. “Nevertheless, the September household survey provides a reason to be a little more optimistic about job opportunities for American workers than we have been in recent months.”

The private sector added jobs for the 31st consecutive month, and the long decline in government work that has been a significant drag on the economy is reversing. Federal, state and local government added 10,000 jobs, and revised figures for the previous two months showed an increase of 63,000 public-sector jobs.

The generally low-paying retail sector added 9,400 jobs last month, while real estate added 7,100, an indication that the housing market is beginning the heal. Restaurants added 15,700 jobs in September, roughly their monthly average over the past year.

Manufacturing employment, which has been a bright spot since the recession ended more than three years ago and a focus of the Obama administration, fell by 16,000 jobs in September after dropping by 22,000 the prior month.