"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJeff SessionsSenate votes to confirm Rosenstein as deputy attorney general C-SPAN to air Trump travel ban arguments live Trump faults DNC in Russian email hacks 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 CollinsOvernight Energy: Lawmakers work toward deal on miners’ benefits Schumer: Senate Russia probe moving too slowly Collins: I'm not working with Freedom Caucus chairman on healthcare 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 BluntDisconnect: Trump, GOP not on same page GOP senator: There will never be full U.S.-Mexico border wall This week: Congress returns to government shutdown fight MORE (Mo.), Scott Brown (Mass.), Thad CochranThad CochranPicking 2018 candidates pits McConnell vs. GOP groups Overnight 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 MORE (Miss.), Susan Collins (Maine), John HoevenJohn HoevenCongress nears deal on help for miners Overnight Energy: Lawmakers work toward deal on miners’ benefits Congress nears deal on help for miners MORE (N.D.), Jerry MoranJerry MoranAt the table: The importance of advocating for ABLE GOP lawmakers lead way in holding town halls Yahoo reveals new details about security MORE (Kansas), Lisa MurkowskiLisa MurkowskiTrump’s Army pick faces tough confirmation fight Republican Sen. Collins considering run for Maine governor in 2018 Alaska senators push bill to allow Arctic drilling MORE (Alaska), Pat RobertsPat RobertsOvernight Energy: Trump to sign orders on offshore drilling, national monuments Watchdog: EPA spending on water pollution campaign was legal Picking 2018 candidates pits McConnell vs. GOP groups 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 ReidWeek ahead: House to revive Yucca Mountain fight Warren builds her brand with 2020 down the road 'Tuesday Group' turncoats must use recess to regroup on ObamaCare 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.