The legislation cleared the Senate hours earlier on a pre-dawn vote of 89-8. White House aides met at the White House to review its progress.
Despite Cantor's remarks, Speaker John Boehner took no public position on the bill as he sought to negotiate a conclusion to the final crisis of a two-year term full of them.
It wasn't the first time that the tea party-infused House Republican majority has rebelled against the party establishment since the GOP took control of the chamber 24 months ago. But with the two-year term set to end Thursday at noon, it was likely the last. And as was true in earlier cases of a threatened default and government shutdown, the brinkmanship came on a matter of economic urgency, leaving the party open to a public backlash if tax increases do take effect on tens of millions.
Economists have warned that without action by Congress, the tax increases and spending cuts that technically took effect with the turn of the new year at midnight could send the economy into recession.
Even with enactment of the legislation, taxes are on the rise for millions.
A 2 percentage point temporary cut in the payroll tax, originally enacted two years ago to stimulate the economy, expired with the end of 2012. Neither Obama nor Republicans have made a significant effort to extend it.
The Senate-passed bill was designed to prevent that while providing for tax increases at upper incomes, as Obama campaigned for in his successful bid for a second term.
It would also prevent an expiration of extended unemployment benefits for an estimated two million jobless, block a 27 percent cut in fees for doctors who treat Medicare patients, stop a $900 pay increase for lawmakers from taking effect in March and head off a threatened spike in milk prices.
At the same time, it would stop $24 billion in spending cuts set to take effect over the next two months, although only about half of that total would be offset with spending reductions elsewhere in the budget.
The non-partisan Congressional Budget Office said the measure would add nearly $4 trillion over a decade to federal deficits, a calculation that assumed taxes would otherwise have risen on taxpayers at all income levels. There was little or no evident concern among Republicans on that point, presumably because of their belief that tax cuts pay for themselves by expanding economic growth and do not cause deficits to rise.
The relative paucity of spending cuts was a sticking point with many House Republicans. Among other items, the extension of unemployment benefits costs $30 billion, and is not offset by savings elsewhere.
"I personally hate it," said Rep. John Campbell of California. "The speaker the day after the election said we would give on taxes and we have. But we wanted spending cuts. This bill has spending increases. Are you kidding me? So we get tax increases and spending increases? Come on."
Others said unhappiness over spending outweighed fears that the financial markets will plunge on Wednesday if the fiscal cliff hasn't been averted.
"There's a concern about the markets, but there's a bigger concern, which is getting this right, which is something we haven't been very good at over the past two years," said Rep. Steve LaTourette of Ohio.
House Democrats met privately with Biden for their review of the measure, and the party's leader, Rep. Nancy Pelosi of California, said afterward that Boehner should permit a vote.
"That is what we expect. That is what the American people deserve," she said.
For all the struggle involved in the legislation, even its passage would merely clear the way for another round of controversy almost as soon as the new Congress convenes.
With the Treasury expected to need an expansion in borrowing authority by early spring, and funding authority for most government programs set to expire in late March, Republicans have made it clear they intend to use those events as leverage with the administration to win savings from Medicare and other government benefit programs.
McConnell said as much moments before the 2 a.m. Tuesday vote in the Senate — two hours after the advertised "cliff" deadline.
"We've taken care of the revenue side of this debate. Now it's time to get serious about reducing Washington's out-of-control spending," he said. "That's a debate the American people want. It's the debate we'll have next. And it's a debate Republicans are ready for."
The 89-8 vote in the Senate was unexpectedly lopsided.
Despite grumbling from liberals that Obama had given way too much in the bargaining, only two Democrats opposed the measure.
Among the Republican supporters were Sen. Pat Toomey of Pennsylvania, an ardent opponent of tax increases, as well as Sen. Ron Johnson of Wisconsin, elected to his seat two years ago with tea party support.
It marked the first time in two decades that Republicans willingly supported higher taxes, in this case on incomes over $400,000 for individuals and $450,000 for couples. Taxes also would rise on estates greater than $5 million in size, and on capital gains and dividend income made by the wealthy.