The fire broke out just after 8 p.m. at the apartments in the 7000 block of Crescent Avenue, according to Capt. Marc Stone of the Orange County Fire Authority.
The fire affected several units and they were deemed a total loss.
The person burned, whose identity was not immediately known, was taken to UC Irvine Medical Center for treatment.
Five others, including two women and three girls ages 14, 7 and 3, were rescued and were taken to area hospitals for smoke inhalation.
The apartment building sustained water and smoke damage. Its power was later turned off.
Stone said some of the displaced were arranging to stay with family members. The Red Cross was stepping in to help as well.
The Orange County Fire Authority and Anaheim Fire Department responded to the scene.
The cause of the fire was not immediately known.