(CNN) -- Positioning herself as a conservative gate-crasher, Christine O'Donnell won more than 53 percent of the vote against U.S. Rep. Mike Castle in Delaware's Republican U.S. Senate primary Tuesday, a contest that showcased the Republican warfare between conservative Tea Party supporters and the more moderate party structures.
In the closing days of the primary campaign, O'Donnell got a boost with an endorsement by former Alaska Gov. Sarah Palin and more than $150,000 in late contributions from the Tea Party Express. Castle, a former Delaware governor who served nine times in the House, got the backing of the national Republican Party.
"We haven't heard from anybody in the Washington party hierarchy. And that's a shame," O'Donnell told CNN's "American Morning" on Wednesday. "I think right now maybe their pride is just hurt a little bit and they're licking their wounds because the so-called experts were discredited this morning.
"But we're hoping to hear from them soon. We will reach out to them. But I'm confident that if they choose not to get behind this race, we will get the support that we need. And we can win in November."
A tiny state populated by about 885,000 people, Delaware has three counties and is bordered by Pennsylvania, Maryland and New Jersey. Because of its size, O'Donnell says, "it's small enough that I can be in every county every week giving the voters an opportunity to get to know me, I can get to know them, and I can personally ask them for their vote in November. And that's exactly what we plan to do."
O'Donnell, who is making her third try for the U.S. Senate, will be running against Democrat Chris Coons, the New Castle County executive, for Vice President Joe Biden's old seat. After O'Donnell won the primary, Coons immediately went on the attack against her, saying he's running against an "ideology" and not a "record." He warns of letting Biden's seat "fall into ultraconservative hands."
"One of Sarah Palin's newest 'Mama Grizzlies,' O'Donnell will fight to roll back a woman's right to choose and lead the charge against stem-cell research, falsely claiming that this ground breaking research exploits women. She has a record of supporting discrimination against gays and lesbians, and pressing for public schools to teach creationism."
"Even more shocking is that despite the fact that she has no plan for putting Delawareans back to work and wants to open our coastlines to more dangerous off-shore drilling risks, she truly believes that she's the right candidate for Delaware. Make no mistake -- Sarah Palin, Jim DeMint, Michele Bachmann and the Tea Party Express will invest to make sure O'Donnell joins them in Washington."
Coons had a picture of O'Donnell on the front page of his website on Wednesday, prompting O'Donnell to say she appreciated the exposure.
"I thank him for introducing me to the Democratic voters I have not met. We have a lot of Democrats working on my team right now. And our message is resonating with voters up and down the state because even the Democrats are frustrated with failed policies coming out of Washington, D.C."
O'Donnell said employment is the top issue in the race, and another major issue is helping veterans.
"I think the biggest concern on everybody's mind is how we're going to get jobs to Delaware. My approach to that is through the private sector. Democrats, independents, and Republicans recognize that these big spending policies -- as in multiple stimulus bills and bailouts coming from Washington -- are not working."
On Tuesday night, O'Donnell thanked her supporters, including Tea Party groups, saying they "rallied everyday Americans outside of the political establishment, [got] them involved and created a grassroots network that made all of this possible."
"Don't ever underestimate the power of we, the people," O'Donnell said to raucous cheers from her supporters. "We, the people, will have our voice heard in Washington, D.C., once again."
Castle thanked supporters in a concession speech Tuesday night.
"The last several weeks have been spirited, shall we say, and the voters of the Republican primary have spoken, and I respect that decision," Castle said.
The GOP establishment is concerned O'Donnell has no chance of winning the seat -- once held by Biden for 26 years. A top Republican official warned Tuesday night that national Republicans will be slow to rally around O'Donnell.
"Until she demonstrates some viability in the polls, we are not going to have any money for her," the official said. "It is now incumbent on Sarah Palin, [U.S. Sen.] Jim DeMint [of South Carolina] and the Tea Party Express to help support her. They got her here. Now make it happen."
O'Donnell told CNN's Jessica Yellin on Tuesday night that she'd "love [establishment Republicans'] support, but they're the same so-called experts who said I couldn't win the primary."
"If we just had that throw-in-the towel mentality every time there was a fight that needed to be fought, our country wouldn't be what it is," O'Donnell said.
"There's a lot of visionaries and leaders ... in this room who believe we can win, and if [establishment Republicans are] too lazy to put in the effort that we need to win, then so be it. We're going to win without them. I'd love their support, but we're going to win without them."
O'Donnell said before Tuesday's primary that "we're breaking up the backroom deals."
"We're restoring the political process back to the hands of the people."
Before the primary, O'Donnell said:
• The Republican Party is behind "false accusations and attack ads" against her, and she accused party operatives of "fighting for not only my opponent's political career but their own political career."
• She believes it's "a shame" that FreedomWorks, the Dick Armey group that backs Tea Party candidates, declined to endorse her but disagrees with the organization's view that she can't win the general election, saying, "No one even thought we could get this far."
• Palin's endorsement "is helping a lot" because "it gave my supporters an extra boost of encouragement." And it helped her, too: "Gov. Palin can relate to the politics of personal destruction, and she's survived them and they didn't get her down."
• And she says she'd love support from one well-known Democrat: "I would love Hillary Clinton's endorsement. When I saw her presidential ads, I said, 'You go, girl!' I'm a Republican, so I probably won't vote for her, but I do admire her."
Democrats were positively gleeful at the prospect of an O'Donnell victory on Tuesday. Many had written off Vice President Joe Biden's old Senate seat as a loss. But with O'Donnell the projected winner, Democrats believe they can hold the seat in November, which could save their majority in the Senate.
CNN's Jessica Yellin and Mark Preston contributed to this report.