"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJeff SessionsRussia is recalling ambassador at center of Trump campaign controversy: report Justice Department developing strategies to shut down ‘sanctuary cities’: report Sally Yates slams Sessions on criminal justice reform MORE (R-Ala.) said just before the Tuesday vote. "Just last August, we agreed to certain debt limits. And we, I believe, have a moral obligation to not mislead the people who elected us when we said we intend to stay by the limits on increasing debt."

On Monday, Sessions argued the bill would require the Treasury to repay $11.4 billion in money that the U.S. Postal Service (USPS) overpaid to its employee retirement system, and would let the USPS defer $23 billion in payments to its retiree health benefit plan. In each case, Sessions said those funds would have to be offset by new government debt, but should instead by offset by spending cuts.

Rep. Susan CollinsSusan CollinsSenate Republicans reluctant to rush vote on healthcare bill GOP senator defends funding Planned Parenthood GOP sen: 'We should not be voting' on healthcare this week MORE (R-Maine), one of the sponsors of the bill, repeated her arguments from Monday that the Congressional Budget Office analysis saying the bill costs $34 billion is "misleading." She said that the funds involved come from the USPS, and that returning them would not amount to any new burden on taxpayers.

"There are no taxpayer dollars authorized by this bill or appropriated by this bill," Collins said.

After brief debate, the Senate voted 62-37 in favor of waiving the point of order, just narrowly meeting the 60-vote threshold. Nine Republicans voted with Democrats to waive the point of order: Roy BluntRoy BluntOvernight Regulation: Senate Banking panel huddles with regulators on bank relief | FCC proposes 2M fine on robocaller | Yellowstone grizzly loses endangered protection Overnight Finance: Big US banks pass Fed stress tests | Senate bill repeals most ObamaCare taxes | Senate expected to pass Russian sanctions bill for second time GOP senator: 'No reason' to try to work with Dems on healthcare MORE (Mo.), Scott Brown (Mass.), Thad CochranThad CochranDefense hawks gird for budget brawl Congressional politics hurts cotton farmers GOP senators dismiss Trump filibuster change MORE (Miss.), Susan Collins (Maine), John HoevenJohn HoevenGOP considers keeping ObamaCare taxes Senators want governors involved in health talks Republicans go to battle over pre-existing conditions MORE (N.D.), Jerry MoranJerry MoranProposal to privatize air traffic control struggles to win over critics Senate panel to reject Trump’s air traffic control plan in aviation bill Senate panel readies must-pass aviation bill MORE (Kansas), Lisa MurkowskiLisa MurkowskiGOP senator defends funding Planned Parenthood GOP at decisive moment on Planned Parenthood GOP Medicaid cuts will be disastrous for millions with Alzheimer’s MORE (Alaska), Pat RobertsPat RobertsThe Hill's Whip List: Senate ObamaCare repeal bill Meet Jay Sekulow, the new face of Trump’s legal team Overnight Healthcare: Senate GOP eyes July vote on health bill MORE (Kansas) and Olympia Snowe (Maine).

With the point of order laid aside, the Senate was expected to immediately consider up to 39 amendments to the postal bill. Senate Majority Leader Harry ReidHarry ReidCharles Koch thanks Harry Reid for helping his book sales Warren cautions Dems against infighting Dems see surge of new candidates MORE (D-Nev.) said he would take up to 10 minutes per vote, but also encouraged some to be held by voice vote to speed up the process.

— This story was updated at 4:18 p.m. to add vote results.