PHOENIX - Former Rep. Gabrielle Giffords' hand-picked Democratic candidate won a special election Tuesday in southern Arizona to finish her term, defeating a Republican who narrowly lost to Giffords in 2010. The race was a hard-fought preview of the broader fall campaign to come.

Both the GOP and the Democrats used the race to hone and test their political arguments for the November elections, when everything from the White House on down will be on the ballot.

Republicans, sensing a chance to capture the former congresswoman's seat, sought to make the contest a referendum on President Barack Obama and his handling of the economy. They argued that Democrat Ron Barber, a former Giffords aide asked by the lawmaker to pursue the seat, would fall in line behind the White House.

Democrats, in turn, played to the senior vote by contending that Republican Jesse Kelly would not protect Medicare and Social Security.

Both candidates promised to run for a full term in the fall, setting up a possible November rematch in a redrawn district that is friendlier to Democrats. Republican voters outnumber Democratic voters by about 26,000 under the current map. That edge will narrow to about 2,000 under redistricting.

Elsewhere Tuesday, Virginia, Maine, Nevada, Arkansas and South Carolina held primary elections - with most of those states choosing Senate nominees - as did North Dakota, where voters decided to let the University of North Dakota scrap its controversial nickname, the Fighting Sioux.

In Virginia, former Sen. George Allen brushed aside three rivals in the Republican Senate primary. Allen's victory set up a November clash with another former Virginia governor, Democrat Tim Kaine, in a campaign closely tied to the presidential race in a state both parties consider vital for victory.

In North Dakota, Rep. Rick Berg defeated businessman Duane Sand in the state's Republican Senate primary. Berg now faces Democrat Heidi Heitkamp in the November race to replace retiring Sen. Kent Conrad. The election is expected to play a critical role in determining which party controls the Senate next year.

The vote concerning the Fighting Sioux nickname came about after the NCAA deemed it hostile and abusive, and placed the university under postseason sanctions. The state's Board of Education is now expected to retire the moniker and American Indian head logo.

In Nevada, Republican Sen. Dean Heller and Democratic Rep. Shelley Berkley easily defeated a slate of political unknowns in their respective primaries. Their fall race will be one of the most competitive in the country.

In Maine, state Sen. Cynthia Dill won the Democratic primary in the race to succeed Republican Sen. Olympia Snowe. Maine Secretary of State Charles Summers won the GOP nomination.

The front-runner, former two-term Gov. Angus King, wasn't on the ballot because he's running as an independent.

No statewide races were part of the Arkansas and South Carolina primaries.

Of all the races Tuesday, the Arizona House race was the most closely watched, partly because of Giffords' absorbing story and partly because holding onto the seat was important for Democrats if they want to regain control of the House.

The party needs big gains in November to grab the majority from Republicans, who now hold a 240-192 advantage with three vacancies, including Giffords' seat.

Republicans, riding high after a decisive victory in Wisconsin's gubernatorial election last week, set their sights on Arizona. A victory would have given party leaders a chance to claim momentum five months before November and fine-tune their plan to link Democratic candidates to Obama, the incumbent at the top of the ticket.

Giffords, 42, resigned in January to concentrate on her recovery from a gunshot wound to her head. She and Barber were injured in the January 2011 shooting rampage outside a Tucson grocery store that killed six people, including a 9-year-old girl and a federal judge. Giffords largely shunned public appearances in the race, but in the closing days she stepped out to help Barber.

Outside groups spent more than $2 million on the race. Barber, 66, had a sizable fundraising lead in late May, but spending from conservative groups helped reduce the Democratic financial edge.

The Arizona 8th is a rare district that is competitive virtually every election. Giffords defeated Kelly by about 4,000 votes in 2010 when the election focused on immigration and when tea partyers rallied to the tough-talking former Marine. Now, the economy and jobs are voters' top concerns.

Kelly, 30, spent the campaign arguing that Barber and Obama are out of touch with people in the district. He called for lower taxes and more energy production as ways to improve the economy. And he said he would roll back federal regulations and environmental protections in an effort to boost oil and gas drilling.

Barber tried to convince voters that he understands their concerns. He frequently talked about building up the solar industry and cutting taxes for the middle class. While Kelly made it clear he would not support any income tax increases, Barber said the wealthy need to "pay their fair share."

The Tucson region is home to a growing population of retirees who rely on Medicare and Social Security. Kelly said in 2010 that privatizing the programs was a "must." He said he would protect Social Security for current seniors but that the program needed to be "phased out." Giffords assailed his comments with great effect, and Democratic groups employed a similar game plan for the special election.

After the victory, Democrats said Barber would build on Giffords' efforts to bring an independent approach to Congress.

"Ron Barber's strong campaign made this a referendum on the Republican plans to drastically cut Medicare and privatize Social Security, while giving massive tax breaks to millionaires, big oil and corporations that ship jobs overseas," said Steve Israel, chairman of the Democratic Congressional Campaign Committee. "The Republican plan lost."

Rep. Pete Sessions, R-Texas, the chairman of the National Republican Congressional Committee, said special elections are unique and predicted that Barber would not fare as well in the fall with Obama leading the ticket.

"No one wanted this election to happen or to see Gabrielle Giffords step down from Congress, but Jesse ran a campaign focused on pro-growth policies that will lead to less government and a strong and vibrant economy," Sessions said.

Freking reported from Washington. Associated Press writers Carson Walker contributed from Tucson and Dave Kolpack contributed from Fargo.