Democrats, in turn, played to the senior vote by contending that Republican Jesse Kelly, who narrowly lost to Giffords in 2010, would not protect Medicare and Social Security.
The victor will complete the remainder of Giffords' term. Both candidates are promising to run for a full term in the fall, setting up a possible November rematch in a redrawn district that is friendlier to Democrats.
Elsewhere Tuesday, Virginia, Maine, Nevada, and South Carolina held primary elections - with most of those states choosing Senate nominees - as did North Dakota, where voters were also deciding whether to go along with the legislature's repeal of a law that required the University of North Dakota to use the Fighting Sioux nickname.
In Virginia, George Allen brushed aside three conservative Republican rivals in the Virginia primary. Allen's victory sets 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 Maine, six Republicans and four Democrats were running for the Senate seat being vacated by GOP Sen. Olympia Snowe. The front-runner, former Gov. Angus King, wasn't on the ballot because he's running as an independent.
In Nevada, Republican Sen. Dean Heller and Democratic U.S. Rep. Shelley Berkley each both were expected to prevail with ease against a slate of political unknowns. Their fall race would be one of the most competitive in the country.
And in North Dakota, U.S. Rep. Rick Berg and businessman Duane Sand were vying for the Republican nomination in the race to replace retiring Democratic Sen. Kent Conrad. Democrat Heidi Heitkamp was unopposed for her party's nomination.
Of all the races Tuesday, the Arizona House race was the most closely watched, partly because of Giffords' absorbing story and partly because holding on to 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 June 5, set their sights on Arizona. A win would give party leaders a chance to claim momentum five months before November and fine-tune their plan to link Democratic candidates to Obama.
Giffords, 42, resigned in January to concentrate on her recovery from a gunshot wound to her head.
Giffords 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 has largely shunned public appearances in the race, but in the closing days, she was stepping out to help Barber.