"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJeff SessionsSanders: 'What do the Russians have on Mr. Trump?' Poll: Trump controversies make him more popular among supporters More than ever, Justice must demand a special prosecutor for Trump-Russia probe 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 CollinsGOP senators pitch alternatives after House pulls ObamaCare repeal bill Five takeaways from Labor pick’s confirmation hearing ObamaCare repeal faces last obstacle before House vote 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 BluntTop Dems prep for future while out of the spotlight Overnight Healthcare: Pressure mounts for changes to GOP ObamaCare bill Pressure mounts for changes to ObamaCare bill MORE (Mo.), Scott Brown (Mass.), Thad CochranThad CochranOvernight Defense: FBI chief confirms Trump campaign, Russia probe | Senators push for Afghan visas | Problems persist at veterans' suicide hotline Senators ask to include visas for Afghans in spending bill Shutdown politics return to the Senate MORE (Miss.), Susan Collins (Maine), John HoevenJohn HoevenSenate panel considers how to fund Trump’s T infrastructure package A guide to the committees: Senate GOP senators unveil bill to give Congress control of consumer bureau budget MORE (N.D.), Jerry MoranJerry MoranGOP lawmakers lead way in holding town halls Yahoo reveals new details about security A guide to the committees: Senate MORE (Kansas), Lisa MurkowskiLisa MurkowskiElle honors 10 at annual 'Women in Washington' event Five takeaways from Labor pick’s confirmation hearing ObamaCare repeal faces last obstacle before House vote MORE (Alaska), Pat RobertsPat RobertsDems mock House GOP over lack of women in healthcare meeting Perdue vows to be chief salesman for US agriculture abroad GOP senator apologizes for mammogram joke 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 ReidThis obscure Senate rule could let VP Mike Pence fully repeal ObamaCare once and for all Sharron Angle to challenge GOP rep in Nevada Fox's Watters asks Trump whom he would fire: Baldwin, Schumer or Zucker 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.