The Arizona Republic said Saturday its analysis of the vote count, which was still under way, showed Barber had enough votes to clinch the 2nd Congressional District race against Republican challenger Martha McSally.
Barber had won an earlier special election to fill Giffords' seat after she was shot during an appearance in Tucson in 2011. Barber -- a former Giffords aide -- was among those wounded in the shooting spree that left six people dead.
KPHO-TV, Phoenix, said Barber ran on a short track record of protecting Medicare and Social Security and keeping middle class taxes low. McSally, a former Air Force pilot, focused on GOP goals of reducing taxes and federal regulations on businesses.
Barber's victory gives Arizona's congressional delegation a Democratic majority for only the second time in 45 years, the Republic said.