Senate leaders Tuesday night resolved an impasse over emergency aid to the jobless that had driven a wedge into Republican ranks and given Democrats a political weapon.

The deal overcame the continued objections of a single senator, Republican Jim Bunning of Kentucky, who for days had held up short-term extensions of unemployment and COBRA benefits over concerns that the bill would increase the federal deficit.

To pacify Bunning, Senate Democrats agreed to allow a floor vote on an amendment to offset the cost of the bill. The amendment did not survive a procedural objection.

The 30-day extension then passed on a 78-19 vote. For at least the next month, hundreds of thousands of people can continue to receive jobless benefits and the federal government can continue to assume a greater share of their COBRA health insurance costs.

The bill also delays a threatened 20% cut in fees for doctors who treat Medicare patients. And it temporarily replenishes the depleted Highway Trust Fund, which will allow suspended construction projects to resume and 2,000 furloughed federal employees to return to work.

Bunning began his one-man blockade Thursday, when he refused to accede to a plan endorsed by both Democratic and Republican Senate leaders that, by unanimous consent, would have approved a short-term extension of several federal programs set to expire Monday. He argued that because the Senate had adopted pay-as-you-go rules, the bill had to identify how its cost would be offset.

“If we can’t find $10 billion to pay for something that all 100 senators support, we’re in deep trouble,” he said Tuesday night. “We are on an unsustainable path, as far as the budget.”

The standoff mushroomed into a significant distraction for Senate Republicans, many of whom were torn between supporting Bunning on principle and being viewed as denying aid to the jobless.

Democrats, for their part, gleefully cast Bunning as the poster child for gridlock. “It’s wrong for one senator to prevent people from getting the help they deserve,” Sen. Barbara Boxer (D-Calif.) said Tuesday.

The agreement to pass the bill may have been driven in part by a Republican moderate, Sen. Susan Collins of Maine. Early Tuesday, she called on Bunning to stand down. He refused.

“The present stalemate is unacceptable,” Collins said in an interview. “This has real-life consequences for people in this country.”

But Bunning drew support from fiscal conservatives nationwide, and some colleagues cheered him on from the sidelines.

“I respect him for the courage he showed,” Sen. Jeff Sessions (R-Ala.) said Monday.

Bunning, 78, had little to lose. The Hall of Fame pitcher had already decided to retire when his term expired early next year, after it became clear that the GOP establishment would not back him for reelection.

Late Tuesday, as the vote approached, Bunning conceded like an athlete, saying he had, in a sense, left it all on the field.

joliphant@latimes.com

Janet Hook and Noam N. Levey in the Washington bureau contributed to this report.