The 89-8 vote, which came about two hours after a year’s end deadline, capped four days of stop-and-start negotiations. The deal would avoid tax hikes on nearly all working Americans while raising taxes on the wealthiest.

Both Bay State Senators John F. Kerry, a Democrat, and Scott Brown, a Republican, backed the compromise. For Brown, it was probably his last significant act as senator before he is replaced by Democrat Elizabeth Warren on Wednesday.

Technically, lawmakers missed the deadline and went over what has been dubbed the fiscal cliff Monday at midnight, triggering tax hikes. But the Senate vote paved the way for the fractious House to follow with its own consideration — perhaps as early as New Year’s Day — giving Congress at least a shot at reversing the tax increases before they have any practical impact on taxpayers.

Representatives in the House will be under intense pressure to also approve it. House Speaker John Boehner was noncommital in a statement released Monday night.

Under terms of the agreement, decade-old income tax cuts would be continued for single taxpayers who earn less than $400,000 a year and $450,000 for couples, and rates would rise on income above those thresholds.

Also for the wealthy, capital gains and dividends taxes would rise to 20 percent from their current 15 percent. And the estate tax formula would be changed to protect large estates from large increases.

On the spending side of the ledger, Democrats would get a delay of two months in $110 billion in cuts; work continued Monday night to satisfy a Republican demand that the delay be offset with about $24 billion in cuts from somewhere else in the budget.

Liberal Democrats raised objections to increasing the thresholds for tax increases to $400,000 and $450,000 from the $200,000 and $250,000 targets promised by President Obama in his reelection campaign, but Vice President Joe Biden helped convince some Democrats in a late-evening meeting that it was important to compromise.

“It’s not that this proposal is regarded as great, or loved in any way, but it is a lot better than going over the cliff,’’ said Senator Charles Schumer, the New York Democrat.

There was no celebrating and no smiles as grim-faced Republicans exited a closed-door meeting after they were briefed on terms late Monday afternoon. The deal represented a watershed moment for the GOP, as senators essentially caved in to demands by Obama and Democrats that income tax rates must be increased on the rich.

For Brown, the vote in favor of the package contradicted pledges he made during his failed reelection campaign not to increase taxes on the wealthy.

“While not perfect, this bi-partisan compromise will protect 99 percent of Americans from a massive tax increase, while giving members the time to continue to work on the very real debt and deficit issues that our nation faces,’’ Brown said in a statement.

Many Republicans wanted the $110 billion in “automatic sequester’’ budget cuts to take effect as scheduled on Wednesday, in exchange for allowing the additional revenues from the tax increases on the wealthy. Those cuts were agreed to in August 2011 as part of the deal to raise the nation’s debt limit. Democrats sought to delay the automatic spending cuts, saying they should be negotiated later.

Obama laid out his opposition to the spending cuts with a staged event near the White House, at the Eisenhower Executive Office Building, where he appeared with a group of supporters and took a few jabs at Congress, even as the sensitive negotiations continued.

“There are some programs that are going to be cut that we are using an ax, instead of a scalpel,’’ said Obama. “That is a piece of business that has to be taken care of.’’

He said Congress had been unable to negotiate a broader, more complex deal, and that it was important that a bad deal that relies upon excessive spending cuts not be “shoved’’ upon the American people.

“If they think that is how we are going to solve this thing, they have got another thing coming,’’ he said. “That is not how it’s going to work.’’