(CNN) -- The Tea Party movement basked in the glow of victory Wednesday after its favorites won primary elections in Delaware and New York the night before over more mainstream Republicans, demonstrating again the clout of the political right.

Now the question is whether the right-wing candidates can also defeat Democratic rivals in November's congressional elections, when the stakes are higher and the full electorate is deciding.

Candidates backed by the Tea Party have won at least eight major GOP nomination fights across the country this year, in Alaska, Delaware, Florida, Kentucky, Nevada, New York, Pennsylvania and Utah. Tea Party candidates have also shown significant strength in numerous other state and local contests.

The results in Delaware and New York highlighted the last major day of primary voting before the upcoming election in just under seven weeks.

Voting in seven states and the District of Columbia included embattled veteran U.S. Rep. Charles Rangel's victory in his New York Democratic primary despite allegations of ethics violations, and D.C. Mayor Adrian Fenty's bid to hold off a major primary challenger.

In addition, former Gov. Robert Ehrlich won the Republican gubernatorial primary in Maryland to set up a rematch against Martin O'Malley, the Democrat who ousted him in 2006.

In Delaware, conservative political commentator Christine O'Donnell easily defeated nine-time U.S. Rep. Mike Castle in the Republican U.S. Senate primary, giving the Tea Party movement another major victory over a candidate backed by the national GOP.

"We the people will have our voice heard in Washington, D.C., once again," a beaming O'Donnell told exuberant supporters at her victory party in Dover.

O'Donnell won more than 53 percent of the vote in the bitter campaign that displayed internal Republican warfare between conservative Tea Party supporters and the more moderate party structures.

Castle was backed by the national Republican Party, while O'Donnell received the endorsement of former Alaska Gov. Sarah Palin as well as $150,000 in late funding from the Tea Party Express.

O'Donnell, running as a Washington outsider, insisted the Republican establishment was trying to drive her out of the race and hand victory to Castle, whom she refers to as "the anointed one."

In response, conservative stalwart Bill Kristol, who fears O'Donnell is incapable of winning the Senate seat in November, said: "I know Sarah Palin. I respect Sarah Palin. And with all due respect -- Christine O'Donnell is no Sarah Palin."

In her victory speech, O'Donnell made a plea for unity, saying: "If those same people who fought against me work just as hard for me, we will win."

Later, she told CNN that she can win without the support of the national Republican Party.

"They don't have a winning track record," O'Donnell said of the national party. "If they're too lazy to put in the effort that we need to win, then, so be it."

The National Republican Senatorial Committee offered its congratulations to O'Donnell immediately after the result was determined.

"We congratulate Christine O'Donnell for her nomination this evening after a hard-fought primary campaign in Delaware," said a statement by Rob Jesmer, the NRSC executive director.

However, a top Republican official told CNN Tuesday night that O'Donnell will have to show she can generate viable support before the national party will give her money.

"It is now incumbent on Sarah Palin, (U.S. Sen.) Jim DeMint and the Tea Party Express to help support her," the official said on condition of not being identified by name. "They got her here. Now make it happen."

O'Donnell told CNN's "American Morning" Wednesday that she has not yet heard from top leaders in the Republican Party hierarchy.

"There are a lot of people rallying behind me who are frustrated that the Republican Party has lost its way. What you see in this race and then especially the attitude after our win is that, you know, the so-called leaders have been proven wrong. They got behind a candidate who didn't even support our party principles, supported the liberals nearly 70 percent of the time some years. And they chose to get behind him because they were taking the easy way out."

O'Donnell will face Democrat Christopher Coons, the New Castle County executive, in November for the seat formerly held by Vice President Joe Biden. This is O'Donnell's third run for the U.S. Senate.

Coons took aim at O'Donnell on his website after her victory, saying "we face an ideology rather than a record."

"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," he said.

O'Donnell, who told "American Morning" that the "biggest concern on everyone's mind is how we're going to get jobs to Delaware."

But Coons says O'Donnell "has no plan for putting Delawareans back to work and wants to open our coastlines to more dangerous off-shore drilling risks."

Commenting on the fact that Coons had a picture of O'Donnell on his website's main page, she said, "I thank him for introducing me to the Democratic voters I have not met."

In New York, conservative Carl Paladino defeated Rick Lazio in the Republican gubernatorial primary to set up a November showdown with Democrat Andrew Cuomo, the son of former Gov. Mario Cuomo. Paladino received Tea Party support in defeating Lazio, who also was supported by some conservative groups.

The New York governor's post has proven hazardous in recent years. Gov. Eliot Spitzer resigned amid a prostitution scandal, and his successor, David Paterson, decided against running for another term due to allegations of wrongdoing involving World Series tickets and a domestic abuse case involving an aide.

In New Hampshire, former state Attorney General Kelly Ayotte squeezed out a narrow victory for establishment Republicans, edging conservative challenger Ovide Lamontagne for the Senate GOP nomination.

Ayotte now advances to the general election, which will determine who succeeds retiring Republican Sen. Judd Gregg.

However, the Republican infighting -- both in New Hampshire and elsewhere -- has raised questions about GOP unity heading into November.

Rangel, meanwhile, received help from former President Bill Clinton in defeating five challengers in the Democratic primary for the House seat he has held for 40 years.

Despite allegations by the House ethics committee that Rangel committed financial wrongdoing and harmed the credibility of Congress, he raised more money than his opponents and easily won the vote in his Harlem district.

The situation was reversed in Washington, where Fenty swept into office in 2006 promising to fix the District of Columbia's struggling schools. But it appeared Wednesday that he won't have the chance to continue that work. CNN affiliate WUSA reported on its website that the mayor's spokesman said he planned to call City Council Chairman Vincent Gray to concede the race. WUSA reported that with 90 percent of the votes counted, Fenty trailed Gray by more than 8,400 votes.

The mayor acknowledged the union opposition to his education reform efforts before the vote.

"We've got an uphill battle because we made tough decisions," Fenty said. "We'll continue to make those tough decisions because they're right for the people. But we're not naive. We know this has cost us a little political popularity that we came into the polls with."

The race was being closely watched far beyond the District of Columbia because the outcome could carry significant implications for the national debate over education reform.

Fenty brought in Michelle Rhee as chancellor of D.C. Public Schools, and she has since become famous for changes that that have become a model of education reform advocated by the Obama administration.

Rhee shut down two dozen schools, fired hundred of educators -- including more than 100 teachers this summer -- for poor performance, and overhauled the teacher evaluation system to include, for the first time, student performance as a measure of success. Local and national teachers unions have fought her efforts.

CNN's Jessica Yellin, Paul Steinhauser, Mark Preston, Kate Bolduan, Kevin Bohn, Mary Snow and Tom Cohen contributed to this report.