The Democratic effort to extend federal benefits for the long-term unemployed got a surprise boost Tuesday as skeptical Republicans in the Senate voted to allow the proposal to advance, with issues of poverty and economic opportunity emerging as a central battleground between the parties.

The reprieve came after President Obama’s outreach to some GOP senators produced the narrowest of victories for the White House.

“We’ve got to make sure this recovery leaves nobody behind,” Obama said after the vote. He stressed that the recession that gripped the nation at the beginning of his first term “was so devastating that there are still a lot of people who are struggling.”

The triumph may be temporary, because the measure still faces big hurdles in the Senate and longer odds of passing the House.

The crux of the negotiations now is the GOP demand for offsetting savings from other portions of the budget. The bill, sponsored by Sens. Jack Reed (D-R.I.) and Dean Heller (R-Nev.), calls for an estimated $6.5 billion to grant an additional three months of benefits for the long-term unemployed.

View Graphic Who are the long-term unemployed?

GOP leaders are increasingly concerned about public perceptions that they are insensitive to those who are still struggling in the slow economic recovery. In a recent memo to rank-and-file Republicans, House GOP leaders urged a show of empathy toward the jobless and advised members to view unemployment as a “personal crisis” for individuals and families.

Seemingly poised for defeat, the legislation instead cleared an early hurdle by the narrowest of margins as six Senate Republicans sided with Democrats to advance it. The sides are now engaged in negotiations over legislation that would allow 1.3 million jobless workers to continue receiving unemployment insurance. The procedural vote in the Senate came as the two parties jockeyed over the political issue of rising income inequality, with Democrats pushing more aid for the jobless and an increased minimum wage. In his speech after the vote, the president called unemployment insurance “a vital economic lifeline” for the millions who are jobless.

Several prominent Republicans, including Sen. Marco Rubio (Fla.) and House Majority Leader Eric Cantor (Va.), plan to tout conservative alternatives to the Democratic proposals and other anti-poverty programs Wednesday as they mark the 50th anniversary of the beginning of President Lyndon B. Johnson’s War on Poverty.

For now, the fight over unemployment has become the main focus of the debate. At the end of December, the emergency laws that extended jobless benefits beyond the traditional 26 weeks expired, forcing about 1.3 million people off the program. More unemployed Americans will lose their benefits as the year progresses and they surpass their states’ normal timelines.

Obama called at least three key Republicans — Sens. Susan Collins (Maine), Mark Kirk (Ill.) and Rob Portman (Ohio) — in the run-up to the vote, signaling that he is willing to discuss other spending cuts.

“When he called, the president did not eliminate the possibility of paying for an extension, but he did not get into how exactly he would do that,” Collins said Tuesday.

White House advisers said that Obama is willing to discuss spending offsets only for a ­longer-term extension of unemployment benefits, not the three-month bill under consideration.

That sets up a delicate negotiation. Of the six Republican senators who voted yes Tuesday — Collins, Portman, Heller, Kelly Ayotte (N.H.), Dan Coats (Ind.) and Lisa Murkowski (Alaska) — five said they were unlikely to support the legislation as it is currently drafted. The six voted with 54 members of the Democratic caucus to approve a motion allowing the measure to move ahead, but Senate Majority Leader Harry M. Reid (D-Nev.) will need to clear a second 60-vote hurdle to bring it to a final vote.

In the House, Speaker John A. Boehner (R-Ohio) took a hard line on what it would take to pass the extension in his chamber; among the GOP’s possible demands are exemptions from Obama’s health-care law and approval for the building of the Keystone XL oil pipeline.

For now, House GOP aides said, Boehner’s leadership team is content to see how the talks play out in the Senate before they consider other options.

Senate Democrats questioned whether Republicans were leading them to a negotiating dead end in which no one could find reductions that would be agreeable to enough Democrats and still win approval from House and Senate Republicans. “We don’t want a Mexican standoff where we put in our pay-for and they put in their pay-for,” Sen. Charles E. Schumer (D-N.Y.) told reporters after the vote.

Democrats would instead like to get approval of the three-month extension for the unemployment benefits and use the intervening time to negotiate offsetting savings for a longer-term extension of the insurance program. But that risks losing votes from key Republicans.

“I don’t buy that argument,” Coats said, suggesting that the 13 previous extensions to the program were too many. He said he will support a new extension only if there are offsetting cuts.

Some Republicans questioned whether Democrats just wanted to make a political point of having the GOP reject unemployment benefits, then criticize them as not understanding the working class.

Senate Minority Leader Mitch McConnell (Ky.) accused Democrats of trying to turn the political focus away from Obama’s health-care law. “I have to admit, I’m a little surprised at the fervor with which the majority is dedicated to reviving the expired emergency unemployment benefits after they ignored the issue all of last year,” he said.

Some Democrats, however, privately suggested that they think extending unemployment benefits is critical to the economic recovery, which is their best path to success in the 2014 midterm elections.

Obama’s personal engagement Tuesday has left some Republicans thinking he is more willing to work out a deal. The GOP senators who spoke with the president were struck by the president’s tone, according to Republican sources who were briefed on the series of calls. The president seemed agitated about the congressional stalemate and was eager to at least consider GOP amendments, even those that would irk Democrats. Obama didn’t make any concrete offers, and White House aides said he made no promises beyond what his advisers have said publicly.

Collins said she told him that after one year of being unemployed, beneficiaries should be required to demonstrate that they are participating in a job-training program. Kirk, who voted no, said he wants Reid to agree to savings.

“If Harry gave us an offset, I’d be a yes,” he said before the vote Tuesday morning.

At that point, there was not enough public support to keep the legislation alive, as Coats and Portman kept their intentions private. Reid privately thought that if they were on the cusp of getting 60 votes, it would materialize, advisers said.

“The rich are getting a lot richer and the poor are getting poorer. The middle class is getting squeezed,” Reid said in a sharply partisan speech before the vote.

Afterward, Portman reserved his right to oppose the final bill, saying his vote Tuesday would allow the Senate to “engage in the debate on how to pay for this and how to make the unemployment insurance work better for Ohioans who are trying to find a job.”

Portman said his office is working on an amendment that would pay for the extension.