Mitt Romney’s commanding performance on the debate stage last week has generated a significant bounce for his presidential candidacy, according to national polls released Monday.

The Republican nominee opened up a 4-point lead over President Barack Obama, 49 percent to 45 percent among likely voters, in the latest national opinion survey by the independent Pew Research Center. In mid-September, Obama led by 8 points, 51 percent to 43 percent, in a survey by Pew, which has tended to show the president with a bigger advantage over Romney than have other major national polls.

Post-debate interviews with 1,201 voters found that the debate had lifted Romney’s standing among a wide range of voter groups. His overall personal image improved, with the percentage of voters holding a favorable opinion of the Republican nominee up 5 points since last month.

Romney also made major gains among two key elements of Obama’s coalition – women and younger voters. The GOP candidate wiped out Obama’s advantage among women voters. Last month, Obama led by 18 points among women, 56 percent to 38 percent; now they are even, 47 percent to 47 percent. And Romney’s image improvement among voters under 30 (he now is viewed favorably by 42 percent of that group, compared with 32 percent in September) was his biggest improvement of any age demographic.

Obama, meantime, suffered broad declines. On jobs, by a margin of 49 percent to 41 percent, voters now say Romney would be better able to improve the nation’s employment situation. That gain for the Republican came even though most of the interviewing for the poll was conducted after Friday’s release of monthly job figures that showed the unemployment rate falling below 8 percent.

By better than 3 to 1, voters said they thought that Romney had done a better job than Obama in the first presidential debate, according to Pew, whose survey had a margin of error of plus or minus 3 percentage points.

That figure is similar to one by the Gallup organization, which also rated Romney the big winner in the debate. A Gallup survey, conducted in the three days following the debate and released Monday, showed a 47-47 deadlock between Obama and Romney among registered voters.

Calling Romney’s debate showing a win of “historic” proportions, Gallup found that even Democratic voters concluded that Romney had done a better job in the debate.

Overall, 72 percent of debate watchers said Romney had won, compared with just 20 percent who said the president did a better job.

That enormous 52-point edge was larger than the previous Gallup debate record, in 1992, when Bill Clinton out-performed President George H.W. Bush by 42 points in a town-hall-style forum that also included third-party candidate Ross Perot.

Gallup cautioned that the recent drop in the jobless rate could “blunt some of Romney’s post-debate momentum.” Gallup’s seven-day tracking poll, a less volatile measure that averaged thousands of interviews between Oct. 1 and Oct. 7, showed Obama with a 5-point lead among registered voters, 50 percent to 45 percent, suggesting that the debate bounce could be ephemeral.

Still to come: the first post-debate polls from battleground states that will decide what is shaping up to be one of the closest presidential re-election votes in decades.

According to Gallup’s Jeffrey Jones, the impact of the debate “was not so strong that it changed the race to the point where Romney emerged as the leader among registered voters.”

But, he added, “even that small movement is significant, given the competitiveness of the race throughout this presidential campaign year and the fact that debates rarely transform presidential election races.”

The Pew poll was based on telephone interviews Oct. 4-7 with 1,511 adults, including 1,201 registered voters and 1,112 likely voters. The margin of sampling error is plus or minus 2.9 percentage points for all respondents, 3.3 percentage points for registered voters and 3.4 percentage points for likely voters.