The draw sees Real two points behind rivals Barcelona, who have won all their fixtures so far and sit on top of LaLiga with 12 points.
Earlier, Barcelona came from behind to maintain their flawless start to the season, defeating Real Sociedad 2-1.
Real are second in the La Liga standings with 10 points after four games.
Visiting coach Ernesto Valverde chose to start his side's arguably least favourite away trip without Philippe Coutinho and Sergio Busquets ahead of a jammed schedule of six games in 20 days.
That gamble appeared to have backfired when Elustondo smashed home following a knockdown from a corner.
The home side had a slew of chances during an entertaining, end-to-end first half, while Real were twice denied by Athletic's impressive 21-year-old goalkeeper Unai Simon.
"In the first half we had more problems for lack of speed in our movements, but we improved in the second half".
Simon made two more outstanding saves to thwart Marco Asensio and Sergio Ramos although Athletic continued to counter, Inaki Williams smashing wide and the Basques being denied a penalty when the ball struck Casemiro on the arm while the Brazilian was on the floor.