Pelosi could not say, however, whether the measure had the backing of most House Democrats.

"Our members are making their decisions now," she said.

Biden joined Democrats for a midday meeting on Capitol Hill seeking to shore up support for the plan.

While Congress technically missed the midnight Dec. 31 deadline to avert the so-called cliff, both sides expressed eagerness to enact a post-facto fix before Americans went back to work and the stock market opened Wednesday.

"This may take a little while but, honestly, I would argue we should vote on it today," said Rep. Tom Cole, R-Okla., who sits on the Budget Committee, early Tuesday. "We know the essential details and I think putting this thing to bed before the markets is important.

"We ought to take this deal right now and we'll live to fight another day, and it is coming very soon on the spending front."

The Senate passed legislation shortly after 2 a.m. that would extend current tax rates on 98 percent of Americans, raise taxes on the wealthiest earners and delay by two months the pending automatic spending cuts to defense and domestic programs, known as the "sequester."

Meanwhile, most senators had already returned home, dismissed early this morning by Majority Leader Sen. Harry Reid.

"I've said all along our most important priority is protecting middle-class Americans, this legislation does that," Senate Majority Leader Harry Reid, D-Nev., said early this morning prior to the vote.

The bill extends Bush-era tax cuts permanently for individuals making less than $400,000 per year and couples making less than $450,000 but allows the top marginal tax rate on incomes above those levels to rise to 39.6 percent.

Capital gains taxes would rise to 20 percent from 15 percent.

The measure would raise the estate tax from 35 to 40 percent for estates larger than $5 million, prevent the alternative minimum tax from hammering millions of middle-class workers and extend unemployment benefits for one year.

Lawmakers also decided at the last minute to use the measure to prevent a $900 pay raise for each member of Congress due to take effect this spring and avert a spike in milk prices by renewing some farm aid.

The steep "sequester" budget cuts scheduled to go into effect with the New Year -- a $1.2 trillion hit to defense and domestic programs -- would be postponed for two months.

ABC News' Arlette Saenz and Matthew Larotonda contributed reporting.