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.
Real are second in the La Liga standings with 10 points after four games.
Earlier, Barcelona came from behind to maintain their ideal start to the season, defeating Real Sociedad 2-1.
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.
It is true that Real Madrid were unable to secure a victory against Athletic Club on Saturday night in the Estadio San Mames.
Real Madrid will aim to continue their ideal start to the season when they travel to San Mamés on Saturday to play Athletic Bilbao - A traditionally tricky trip for the European champions that last season ended in a goalless draw.
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.
The visitors finally beat the youngster when Isco nodded home a cross from Gareth Bale in the 63rd minute as the visitors improved after bringing on midfield anchor Casemiro at the start of the second half.
"When you can get three points, it's not good to just take one".
Atletico had dominated proceedings but were frequently thwarted by Eibar's Serbian keeper Marko Dmitrovic and it took the unlikely figure of academy graduate Garces, 19, to pull them level in the 94th minute, thumping the ball home after controlling a cut-back from Angel Correa.