In the latest test of the country’s politics of polarization, the middle ground held on.

Senator Chafee, a Republican of Rhode Island, said his victory Tuesday against a conservative challenger sent a message across the nation that moderate Republicans remained “alive and kicking.” “Partisan politics must not prevail,” he said.

The Rhode Island Senate primary was the most closely watched on the last big day of primaries before the November elections, with races also in Arizona, Delaware, Maryland, Minnesota, New Hampshire, New York, Vermont, Wisconsin, and the District of Columbia.

Mr. Chafee’s primary win over Cranston Mayor Steve Laffey was notable in a year that saw moderates from each party lose primaries to hard-line candidates. He thanked Democratic-leaning independents for supporting him while he got backing from the Bush administration — with his seat critical to Republicans trying to hold a Senate majority — even after he bucked them on Iraq, taxes, and the environment.

White House press secretary Tony Snow congratulated Mr. Chafee, saying on ABC’s “Good Morning America” yesterday that, while Mr. Chafee and President Bush disagree on some policies, “so do conservatives from time to time.” “The most important thing is he’s a loyal Republican. We’re glad to have him aboard,” Mr. Snow said.

Another test of conservatives versus moderates for an Arizona House seat brought a different outcome, with the potential to play a part in the larger fight for control of Congress.

Conservative former lawmaker Randy Graf, who ran hard against illegal immigration in the district that stretches from Tucson to the Mexican border, beat moderate state Rep. Steve Huffman 43% to 37%.

National GOP leaders had angered local Republicans when they jumped into the race to support Mr. Huffman for the seat left open by retiring GOP Rep. Jim Kolbe, worried that Mr. Graf might be too conservative to beat Democratic candidate a former state legislator, Gabrielle Giffords.

“We stayed on the issue, kept our campaign on the up-and-up, and talked about the issues we’ve talked about for years, and the voters appreciated that,” Mr. Graf said.

Each party has struggled this year with intra-party challenges. Senator Lieberman of Connecticut lost the Democratic primary to anti-war candidate Ned Lamont, though the incumbent is still running as an independent. Rep. Joe Schwarz, a moderate Republican of Michigan, targeted by the anti-tax Club for Growth, was defeated.

Two other incumbents lost primaries, though both had grown unpopular for their personal behavior — Governor Murkowski, a Republican of Alaska, and Rep. Cynthia McKinney, a Democrat of Georgia.

In Rhode Island, the importance of holding onto a GOP Senate seat brought Laura Bush and the GOP establishment to campaign for Mr. Chafee — even though he was the only Republican to vote against the resolution to use force against Iraq, and he opposed the president’s tax cuts. Mr. Chafee did not even vote for Mr. Bush in 2004 — instead writing in the name of Mr. Bush’s father, former President George H.W. Bush.

Mr. Chafee said he did not expect the current president to campaign on his behalf. “His current approval ratings would not be helpful in this Democratic state,” Mr. Chafee said.

With 99% of precincts reporting, Mr. Chafee had 34,042 votes, or 54%, to Laffey’s 29,431 votes, or 46%.

Polls show Mr. Chafee will still face a tough contest against a former attorney general, Democratic nominee Sheldon Whitehouse. But if Mr. Chafee had lost, polls showed that Mr. Whitehouse was almost assured a victory. Democrats hope to build on dismay with Mr. Bush to capture majorities in Congress, and they need six Senate seats.

Mr. Chafee, 53, was appointed to the Senate in 1999 after his father, John Chafee, a governor and senator, died in office. It was the younger Mr. Chafee’s opposition to Mr. Bush’s tax cuts that spurred the anti-tax Club for Growth to back Mr. Laffey.

In New York, front-running Democrats swept aside primary challengers — Senator Clinton trounced an anti-war candidate in her re-election bid, Attorney General Eliot Spitzer crushed his opposition for the Democratic nod for governor, and Andrew Cuomo easily won the party nomination for attorney general.

In New York, Ms. Clinton beat challenger Jonathan Tasini with more than 80% of the vote. She will face former Yonkers Mayor John Spencer.

Mr. Spitzer defeated Nassau County Executive Tom Suozzi with more than 80% of the vote. He will face a former legislative leader, GOP candidate John Faso, in the fall. Former federal Housing Secretary Andrew Cuomo — son of Governor Cuomo — defeated the former New York City Public Advocate, Mark Green, to win the Democratic nomination for attorney general.

And City Council member Yvette Clarke, who is black, prevailed in a racially charged congressional primary in the heart of Brooklyn, beating white City Councilman David Yassky.