Real had the best chances of a poor game in San Sebastian, but Sergio Canales fired just wide and substitute Jonathas couldn't bundle the ball home in a goalmouth scramble two minutes from time.
A point takes Moyes's side up to 12th, while a draw was enough to move Athletic out of the relegation zone.
Elsewhere, Deportivo la Coruna moved up to sixth as they eased past 10-man Espanyol 3-0 thanks to two goals from Lucas Perez.
Real Betis came from behind to win 2-1 at Sporting Gijon as Joaquin grabbed his first goal since returning to his boyhood club before Ruben Castro struck a spectacular winner.
Getafe also moved out of the bottom three with three goals in the last 10 minutes to beat Levante 3-0.