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 is 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 recall 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.
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.
Democratic officials were thrilled that Barber won a district that President George W. Bush carried with 54 percent of the vote in 2004 and that John McCain carried with 53 percent of the vote when he ran against Obama.
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.
Freking reported from Washington. Associated Press writers Carson Walker contributed from Tucson and Dave Kolpack contributed from Fargo.