• The Social Security Fairness Act passed by a vote of 76-20 in the early hours of Saturday.
• The act eliminates provisions that affect an estimated 3 million individuals.
• The measure was passed in the Senate after a key amendment, sponsored by Sen. Rand Paul, R-Kentucky, was defeated.

As Congress managed to avoid a government shutdown, the Senate also passed a key bill that will increase Social Security benefits for some public workers.

The Social Security Fairness Act, which passed by a vote of 76-20 in the early hours of Saturday, will eliminate Social Security provisions known as the Windfall Elimination Provision, or WEP, and Government Pension Offset, or GPO, that have been in place for decades.

The WEP reduces Social Security benefits for individuals who receive pension or disability benefits from employment where they did not pay Social Security payroll taxes. The GPO reduces Social Security for spouses, widows and widowers who also receive their own government pension income. Together, the provisions affect an estimated 3 million individuals.

The bill had enthusiastic support from organizations representing teachers, firefighters, police and other government workers who are affected by the benefit reductions.

"You shouldn't penalize people for income outside of a system when you've paid into it and earn that benefit," said John Hatton, vice president of policy and programs at the National Active and Retired Federal Employees Association, ahead of the vote. "It's been 40 years trying to get this repealed."

The bill had received overwhelming bipartisan support. The Social Security Fairness Act was passed by the House with a 327 majority in November.

The measure was passed in the Senate after a key amendment, sponsored by Sen. Rand Paul, R-Kentucky, was defeated.

Amendments may include raising the retirement age

The Social Security Fairness Act would cost an estimated $196 billion over 10 years, according to the Congressional Budget Office.

Those additional costs come as the trust funds Social Security relies on to help pay benefits already face looming depletion dates. Social Security's trustees have projected the program's trust fund used to pay retirement benefits may be depleted in nine years, when just 79% of benefits may be payable.

Some senators who opposed the Social Security Fairness Act had expressed concerns about the pressures the additional costs would put on the program.

Sen. Paul, who earlier this week voted against moving the current version of the bill forward in the Senate, proposed an amendment to offset those costs by gradually raising the retirement age to 70 while also adjusting for life expectancy. Social Security's full retirement age — when beneficiaries receive 100% of the benefits they've earned — is currently age 67 for individuals born in 1960 or later.

"It is absurd to entertain a proposal that would make Social Security both less fair and financially weaker," Paul said at the time. "To undo the damage made by this legislation, my amendment to gradually raise the retirement age to reflect current life expectancies will strengthen Social Security by providing almost $400 billion in savings."

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Social Security advocacy groups have pushed for larger comprehensive Social Security reform that would use tax increases to pay for making benefits more generous.

"We want to help in making this happen, but our preference was for it to be part of a much larger Social Security reform," said Dan Adcock, director of government relations and policy at the National Committee to Preserve Social Security and Medicare, before the vote.

— CNBC's Katrina Bishop contributed to this report.