Ander Capa equalised for Athletic at the second attempt after he had a shot saved by Thibaut Courtois.
Benzema scored twice to win the game for Real, heading in and then drilling an injury-time second.
That clinching goal came moments after Courtois denied Mikel Vesga an equaliser with a fine save.
Real have now won four games in a row in all competitions as they continue to turn their season around.
Real Sociedad, Atletico Madrid, and Real Madrid are all on 26 points at the top of La Liga.
Barcelona, who are nine points behind the trio, host Real Sociedad on Wednesday (20:00 GMT).