WASHINGTON — The Senate broke a stalemate on Tuesday over extending unemployment benefits for Americans who have been out of work for six months or more, voting to override Republican objections that the bill’s costs would add to the federal deficit.

On a 60-to-40 vote, the Democratic-led Senate agreed to cut off debate on the $34 billion plan to distribute added unemployment compensation through November for those who have exhausted their standard 26 weeks of aid.

The 60 yeas were the minimum needed to overcome the threat of a filibuster and advance the bill to a final vote, expected later Wednesday, when it is all but certain to pass. Two Republicans, Senators Susan Collins and Olympia J. Snowe of Maine, joined 56 Democrats and two independents in voting for the legislation; 39 Republicans and one Democrat, Senator Ben Nelson of Nebraska, opposed it.

An estimated 2 million Americans have seen their benefits run out over the past two months while the legislation has been stalled in the partisan impasse.

“Finally, finally, finally,” said Senator Barbara A. Mikulski, Democrat of Maryland. She called the unemployment insurance program a social compact with American workers that meant “when you hit a speed bump and have to be laid off through no fault of your own, there will be a safety net so that you do clearly fall.”

Republicans said they backed the idea of extending benefits, but were determined to prevent the costs from being piled onto the mounting deficit.

“We believe the federal debt has grown to an alarming level, where it is threatening the future of our children and grandchildren,” said Senator Lamar Alexander of Tennessee, the No. 3 Republican in the Senate.

After the Senate completes its final vote on the measure, the House must still act on it, a vote that is could come shortly after expected Senate passage on Wednesday. President Obama would then quickly sign the bill into law at the White House, freeing the aid.

The Senate action came just minutes after Carte Goodwin was sworn in as the new Democratic senator from West Virginia, replacing the late Robert C. Byrd. While the seat was vacant, Democrats lacked the votes to overcome the Republican filibuster.

At age 36, Mr. Goodwin, a former legal adviser to Gov. Joe Manchin III, becomes the youngest member of the Senate; Mr. Byrd, at 92, had been the oldest.

Both parties expect the fight over the unemployment extension to figure in midterm election campaigns in the fall. Republicans say they believe their stand will strengthen them with voters worried about the rising deficit. Democrats will criticize Republicans’ willingness to set aside deficit concerns when they pushed through tax cuts for the affluent, but clearly when unemployment pay for ordinary workers was at stake.