Goodwin was the crucial 60th senator to defeat a Republican filibuster that has led to a lapse in benefits for 2.5 million people.
The Senate is expected to vote on Wednesday on the bill itself, then send it to the House.
A battle has raged for months over whether jobless benefits should be financed with additional federal debt as Democrats want or through cuts to other government programs as most Republicans insist.
The Senate extension bill would extend the deadline to file for extended unemployment benefits through Nov. 30, and would retroactively pay out claims to those who saw their benefits expire in May, according to the House and Senate versions of the bill.
To date, the Senate bill does not include a COBRA health insurance subsidy or Tier 5 unemployment extension that would provide additional weeks of unemployment for the "99ers" -- the unemployed workers who have exhausted all unemployment benefits.
In short, the bill will allow American workers who have lost their jobs to receive up to 99 weeks of federal unemployment benefits through November, rather than the usual 26 weeks.
The extension is retroactive, so unemployment coverage will resume for the 98,000 Ohioans who already exhausted their benefits as a result of the delay in Congressional action.
The extension of unemployment provides much needed relief for the hundreds of thousands of Ohioans without a job and an important boost to Ohio's economy.
Economists have shown that programs like Unemployment Insurance, where recipients are especially likely to spend any extra funds right away within their communities, are the most effective at stimulating the economy.
Through the end of May, $1.38 billion came to Ohio's workers from the extra weeks of unemployment.
If the House passes the bill Wednesday or Thursday, President Barack Obama is likely to sign it into law by week's end.
After the Tuesday vote, Obama released this statement:
"Today marks an important step toward passing the unemployment insurance extension which is critical to millions of Americans fighting to find a job, put food on the table and make ends meet during this tough economic time."
"After a partisan minority blocked this critical aid to our nation's families three separate times, the Senate has moved forward on restoring benefits to the 2.5 million Americans whose livelihood has been held hostage by obstruction and game playing over the past weeks."
"I will continue to fight for economic policies that will lead us out of this mess, and press Congress to act on more proposals to create new American jobs and strengthen our recovery, including a small business jobs bill, aid for struggling states to prevent layoffs, and tax cuts for middle class families."
"Americans who are struggling to find a job and get back on their feet deserve more than the same political game-playing and failed policies that helped cause this recession. And I thank the members of the Senate who stood on the side of these working families today."
Goodwin voted with 57 other Democrats and two Republicans -- Sens. Susan Collins and Olympia Snowe of Maine -- in favor of "cloture," which will end the filibuster and allow the Senate's unemployment benefits extension bill to move forward towards final passage.
(Cloture is the only procedure by which the Senate can vote to place a time limit on consideration of a bill or other matter and, thereby, overcome a filibuster.)
After they pass cloture, the Senate will still have to take one more vote on the bill -- within 30 hours of passing cloture -- before they can send it over to the House of Representatives for the final vote -- now expected on Wednesday -- before it is sent to President Barack Obama for his signature.
Senate Majority leader Harry Reid expects the vote after cloture will be the final passage of the extension bill.
After Congress failed to extend unemployment benefits in June, the U.S. House passed its "stand-alone" unemployment insurance extension bill -- H.R. 5618 -- by a 270-153 margin on July 1.
H.R. 5618 extends the filing deadline for extended unemployment insurance benefits until Nov. 30, 2010.
Why was the Senate filibustering?
Republicans and Nebraska Democrat Ben Nelson have been preventing a final vote on the Senate bill because of its $33 billion cost.
Reid, in a statement last week, said the GOP filibuster, which has prevented more than 2.1 million people from receiving checks, is designed to crater the economy.
"They're betting on failure. They think the worse the economy is come November, the better they're going to do election-wise," said Reid. "Almost two million people who are long-term unemployed. These are not numbers. They are people."
Congress allowed extended benefits for people who've been out of work for longer than six months to lapse at the beginning of June.
Since then, Senate Democrats have repeatedly failed to muster 60 votes to overcome the deficit reduction demands of the Sen. Ben Nelson (D-Neb.) and the Republican party.