Story highlights House approves the deal late Tuesday night

The vote was an up-or-down vote on the Senate compromise

Lawmakers were under a tight deadline; A new Congress is set to be sworn in Thursday

The House of Representatives voted Tuesday night to approve a Senate bill to avert a feared fiscal cliff.

The measure that sought to maintain tax cuts for most Americans but increase rates on the wealthy passed the Democratic-led Senate overwhelmingly early in the day.

There was discussion about amending the Senate bill by adding spending cuts, but in the end, House lawmakers voted on the bill as written -- a so-called up or down vote.

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The legislation would raise roughly $600 billion in new revenues over 10 years, according to various estimates.

"I'd say let's take the Senate deal, fight another day," Rep. Tom Cole, R-Oklahoma, told CNN before the House vote. He predicted the House would pass the bill with a "pretty strong bipartisan majority."

"I'm a very reluctant yes," said Rep. Nan Hayworth, an outgoing Republican representative from New York.

"This is the best we can do given the Senate and the White House sentiment at this point in time, and it is at least a partial victory for the American people," she said. "I'll take that at this point."

The timing of the vote was crucial, as a new Congress is set to be sworn in Thursday.

The legislation averted much of the fiscal cliff's negative near-term economic impact by extending the Bush-era tax cuts for the majority of Americans. It also extends long-term unemployment benefits that were set to expire.

Had the House not acted, and the tax cuts enacted last decade expired fully, broad tax increases would have kicked in, as would $110 billion in automatic cuts to domestic and military spending.