Barber -- also seriously wounded in the Jan. 8, 2011, shooting spree at a Giffords meet-and-greet political event outside a grocery store in Tucson -- defeated Republican Jesse Kelly 53 percent to 45 percent, unofficial results indicated.
Kelly, a former Marine and Iraq War veteran, conceded the race for the conservative-leaning southeastern Arizona district to Barber, 66, in a speech to Tucson supporters late Tuesday.
Barber told supporters: "Just about 10 minutes ago I received a call from Jesse Kelly. It was a most gracious call. And Jesse said, 'Congratulations on your victory, and I know you will make a great congressman for Arizona.' How about that?"
Giffords, who resigned Jan. 25 to focus on her recovery from a bullet wound to the head, supported Barber and campaigned on his behalf, but played only a small role in the campaign.
Barber will complete the remainder of Giffords' term. Both candidates promised to run for a full term in the fall, setting up a possible November rematch in a slightly redrawn congressional district.
Fox News Channel said the new district lines could be friendlier to Democrats. CNN said the race would still be competitive.
House Minority Leader Nancy Pelosi, D-Calif., cheered the outcome of the final congressional special election before November's general election.
"Congressman-elect Barber follows in the footsteps of our extraordinary colleague, Congresswoman Gabby Giffords -- and he has enormous shoes to fill," Pelosi said in a statement.
"Gabby left a legacy of strength, resolve, and independence in the House," the statement said. "We look forward to Ron Barber continuing in that same tradition, and we congratulate him on a well-deserved victory."
National Republican Congressional Committee Chairman Pete Sessions, R-Texas, said in a statement, "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.
"It is clear that Ron Barber knew that voters in this district would never accept his true positions on President [Barack] Obama's agenda which have made a bad economy worse in this state," Sessions' statement said. "That explains why he did his best to conceal his support for so much of that agenda. Barber will not have that advantage in November when he will be on the ballot with President Obama, nor will any of his House Democrat colleagues."