Amy Kremer, leader of the group Tea Party Express, said Thursday she is "outraged" over the deal, which was struck between Senate Republican Leader Mitch McConnell and Vice President Joe Biden. It allows tax rates on wealthy earners - families earning over $450,000 and individuals earning over $400,000 - expire while extending the tax breaks passed under President George W. Bush for earners below those thresholds.
"We elected the House of Representatives - we have the majority, the Republicans have the majority there. We still control part of the government and the House controls the purse strings," Kremer said on CNN.
"At some point we're going to have to come together and do what's right for America, but do the Democrats have a mandate to go and just do whatever and just spend into oblivion? No," she said. "And that's evident because the Republicans still control the House."
In the November House elections, Republicans posted a net loss of 8 seats.
Another conservative group, Tea Party Patriots, urged the new House to select their speaker through a secret ballot "so members can vote their conscience without fear of retribution from leadership."
"Through this 'fiscal cliff' crisis Congress has demonstrated once again its incapability of actually cutting government spending and preventing tax hikes," the group's head, Jenny Beth Martin, said in a statement. "Republicans who stood on principle, and did not cave to political pressure instead should not be impeded from voting for a speaker with whom they have confidence will help take the country in the right direction."
While negotiations between House Speaker John Boehner and the White House were often at a stalemate, Boehner did offer a compromise on taxes, laying out a plan which included tax increases for earners of over $1 million in exchange for GOP-backed elements, including a change in entitlement benefit calculations.
Rep. Steve LaTourette, R-Ohio, who is seen as a moderate in the GOP, said on CNN tea party conservatives have held "a feeling after the 2010 election that leaders of both parties weren't doing enough to rein in the deficit and the debt. ...
"So as a result, they have laid obstacles in front of the speaker for the last two years," he said, including the "fight on the fiscal cliff" and over a bill with appropriations for the states impacted by Superstorm Sandy.
After the vote, Kremer posted on Twitter, "I'm extremely disgusted with what happened in the House tonight. There will be consequences."
It passed the Senate 89-8 with a majority of Republicans - 42 of them - casting yes votes.
"I care more about my country than I do about a 20-year-old pledge," Chambliss said in a post-election November interview with WMAZ, a CNN affiliate. "If we do it his way, then we'll continue in debt and I just have a disagreement with him about that."
Kremer has promised the two-term senator will face a primary challenge in 2014.
"I live in the state of Georgia and Saxby Chambliss is going to be 'primaried,' our own senator," she said in the CNN interview. "It's unacceptable to have somebody who votes with the Democrats more than they do with the conservatives, and he has proven time and time again he is all about the spending. And we're a red state, we deserve a conservative senator."
Ted Cruz, a Texas Republican and tea party favorite, was elected this year to the seat held by retiring Sen. Kay Bailey Hutchison. He said Thursday morning that he was disappointed with the agreement.
"I think it was a lousy deal," he said. "I think it raised taxes by $620 billion, which is going to hurt the economy, it's gonna kill jobs. And to combine that not with spending cuts but with spending increases, $330 billion in additional spending - that doesn't solve the problems we've got."
Besides raising revenue through increased taxes on wealthy earners and limiting tax exemptions for the wealthy, the law spends by extending long-term unemployment insurance and averting a scheduled pay cut to doctors who provide Medicare services. The Congressional Budget Office said the deal would spend $332 billion over ten years.
– CNN's Jim Acosta and Rachel Streitfeld contributed to this report