But not before Republicans accused Democrats of playing politics.
"It is transparent that this is a political exercise, not a real effort to try to fix the problem," Sen. John Cornyn, R-Texas, said in a protest followed immediately by Majority Leader Harry Reid's agreement to delay the vote.
It was unclear whether the delay would affect the fate of the bill.
Democratic supporters of the three-month extension of jobless benefits need 60 votes to advance the White House-backed bill, and their chances hinge on securing backing from at least four Republicans in addition to Sen. Dean Heller of high-unemployment Nevada.
Sen. Susan Collins of Maine told reporters she would vote to advance the bill, in the hope that Republicans would have a chance to offer changes that would offset the cost and prevent deficits from rising.
(Read more: Six years post recession, a tale of two Americas)
Other Republicans weren't as optimistic.
Sen. Bob Corker of Tennessee said he would vote the other way. "Unfortunately, this bill is being jammed through, has not been considered in committee and will not be able to be amended on the floor," he said. "Spending $6.5 billion in three months without trying to find ways to pay for it or improve the underlying policy is irresponsible and takes us in the wrong direction," he added.
As drafted, the bill would restore between 14 weeks and 47 weeks of benefits averaging $256 weekly to an estimated 1.3 million long-term jobless who were affected when the program expired Dec. 28. Without action by Congress, thousands more each week would feel the impact as their state-funded benefits expire, generally after 26 weeks.
"These are people who want to work, but they need some help," Sen. Jack Reed, D-R.I., said of the men and women who have been out of work longer than 26 weeks. He said many are middle-class, middle-aged people who never thought they would wind up in the situation in which they find themselves.
Reid said that as the unemployed spend the funds they receive, the overall economy grows by $1.50 for every $1 in benefits.