Barber and Kelly will face off in the special general election in June, but the two opponents have already started the war of words.
Barber said Kelly has "the wrong priorities" for southern Arizona's middle class families and seniors.
"He supports a radical tax plan that would increase taxes by 23 percent for middle class families. He's said that Medicare should be eliminated and Social Security should be phased out," Barber said. "I am running to rebuild southern Arizona's middle class... I think now the race is engaged."
Kelly said Barber is wrong and his focus is on cutting taxes, creating more jobs by building a stronger economy and lowering gas prices by opening up more federal lands to exploration.
He's proposed a 10-percent flat tax and said "every single American should pay the exact same tax rate."
"I think a majority of the voters like our plans," Kelly said after his win Tuesday night.
Kelly had about 38 percent of the vote and the margin was enough for The Associated Press to declare him the winner in the race. He beat three other GOP candidates — state Sen. Frank Antenori, retired U.S. Air Force pilot and political newcomer Martha McSally, and Tucson businessman and longtime sports broadcaster Dave Sitton.
Antenori and McSally both conceded before final results were available.
Giffords tapped Barber as her preferred replacement, and he drew strong financial support and no challengers in the Democratic primary.
Kelly is a businessman and tea party favorite who nearly unseated Giffords in 2010.
The winner of the June 12 special general election will hold the seat until Giffords' term expires at the end of 2012. The race for the full term for the renumbered District 2 kicks off with an August primary.
Giffords resigned from her seat on Jan. 25, slightly more than a year after she was shot at a constituent meet-and-greet in Tucson. Six people were killed and 13 others were wounded including Barber and Giffords.
The 8th District spans parts of Tucson and its suburbs, some of Pinal and Santa Cruz counties and all of Cochise County.
All four Republicans in the race embraced strong conservative issues like enhanced border security before immigration reform, repeal of federal health care legislation and cuts to the size of federal government and of federal regulations. That may be an issue for Kelly since Barber embraces Giffords' moderate Democratic positions, which can be a draw to the independents who decide the general elections in the district.
The 8th District has nearly 425,000 registered voters with 159,000 Republicans, 134,000 Democrats and 128,000 independents.
Copyright 2012 The Associated Press.
