A bipartisan agreement to ease sharp spending cuts known as the sequester easily cleared its last major hurdle Tuesday as 67 senators voted to advance the measure in hopes of ending nearly three years of political gridlock over the federal budget.

Despite the opposition of conservative advocacy groups and late-breaking concerns about proposed reductions to military pensions, 12 Republicans joined all 55 members of the Senate Democratic caucus in voting to proceed to a final vote, expected to occur Wednesday evening.

Thirty-three Republicans voted to block the deal, a modest package of fees and spending cuts designed to replace about half the sequester cuts in the current fiscal year and avert another government shutdown. However, it would have little impact on the larger problems at the heart of Washington’s budget battles, and even those who voted to support it were hardly enthusiastic.

“There’s a heavy sigh going on in our caucus right now, because people like me are looking at this and saying, ‘Is it a good deal? No. But is it a deal? Yeah,’ ” said Sen. Lisa Murkowski (R-Alaska), who voted to advance the legislation. “I have tried to look at this on balance. And on balance, the benefit of having a deal is better than no deal.”

The agreement was brokered on the heels of a 16-day government shutdown in October that riled taxpayers and sent approval ratings for congressional Republicans plummeting in public opinion polls. After nearly two months of talks, Senate Budget Committee Chairman Patty Murray (D-Wash.) and House Budget Committee Chairman Paul Ryan (R-Wis.) last week announced the deal, which would ease the impact of the sequester without raising taxes or cutting popular retirement programs.

Washington Post economic columnist Neil Irwin tells you everything you need to know about the new budget deal — in two minutes. (Sarah Parnass/In Play/The Washington Post)

“We have lurched from one crisis to another, from one fiscal cliff to the next,” Murray said before the vote. “I am hopeful this deal can be the first of many bipartisan deals that can rebuild some of the trust.”

Although the agreement sailed through the House on a landslide vote of 332 to 94, problems quickly cropped up in the Senate. Potential GOP presidential hopefuls, including Sens. Marco Rubio (Fla.), Rand Paul (Ky.) and Ted Cruz (Tex.), opposed it because it would increase spending now in exchange for the promise of budget cuts later.

A number of Republican incumbents facing primary challenges from the right also chimed in, including Sen. Thad Cochran (Miss.), whose GOP opponent, Chris McDaniel, on Monday urged Cochran to oppose the “disastrous,” “deficit-increasing legislation.”

And Senate Minority Leader Mitch McConnell (R-Ky.) voted no, citing his long-standing opposition to any agreement that would weaken the sequester, which was part of a historic $2.1 trillion deficit-reduction deal McConnell negotiated in the summer of 2011 with Vice President Biden. McConnell faces serious challenges from the left and the right in his 2014 reelection campaign, and he has touted the sequester agreement as reducing the size of government and ending an economy-shaking fight over the debt limit.

On Tuesday, however, concerns about cuts to military pensions took center stage. The deal calls for $85 billion in total savings over the next decade, including $6 billion from reducing cost-of-living adjustments by 1 percent for military retirees younger than 62.

According to House budget aides, the proposal would reduce lifetime retirement pay by about 6 percent for a man who enlisted at age 18 and retired at age 38 as a sergeant first class in the Army — leaving him with about $1.626 million in lifetime retirement pay instead of $1.734 million.

Sen. Kelly Ayotte (R-N.H.) called the provision “a dealbreaker” and urged her colleagues to find an alternative. “We could quickly find $6 billion that would not be taken from the backs of our men and women in uniform,” she said.

That prompted her home-state colleague, Sen. Jeanne Shaheen (D), to propose replacing the pension cuts by closing a variety of corporate tax loopholes — an idea Republicans roundly ignored.

Meanwhile, Sen. John McCain (R-Ariz.), one of the few senators who served in the military, dismissed the outcry, noting that even high-ranking Pentagon officials have acknowledged the need to rein in the cost of military benefits, from pensions awarded after only 20 years of service to the military health program known as TriCare.

“We have to be careful, because this is an all-volunteer force,” he said. “But when people say, ‘Well, they joined up because of TriCare’ — I do not know of a single 18-year-old, including my own son, who joined the Marines at 18. I said, ‘Jimmy, do you to this day know what TriCare is?’ And he said, ‘Hell, no, Dad. What is TriCare?’ ”

McCain voted to advance the budget agreement, citing the need to avoid another government shutdown and protect the Pentagon from fresh sequester cuts set to hit in January. Other Republicans voting for the measure were Murkowski, Lamar Alexander (Tenn.), Roy Blunt (Mo.), Saxby Chambliss (Ga.), Susan Collins (Maine), Jeff Flake (Ariz.), Orrin G. Hatch (Utah), John Hoeven (N.D.), Johnny Isakson (Ga.), Ron Johnson (Wis.) and Rob Portman (Ohio).