A highly entertaining contest saw both teams have a number of opportunities to score, but neither could find the back of the net, with the points ultimately being shared in the Spanish capital.
The result has left Carlo Ancelotti's five points from Barcelona on the same number of matches (18), while Real Sociedad stay third, three points behind their opponents in this match.
The visitors came close to taking the lead in the sixth minute when Aihen Munoz delivered a brilliant cross towards Alexander Sorloth, but the forward just could not make contact under pressure.
Vinicius Junior then placed one just wide of the away side's goal in the 14th minute, with Real Madrid starting to threaten after feeling their way into the contest at Bernabeu.
Real Sociedad continued to look dangerous, though, and Asier Illarramendi hit the base of the post in the 17th minute after Toni Kroos could only half-clear another dangerous ball from Munoz.
Alex Remiro then made two smart saves late in the first half, first keeping out a drive from Kroos before managing to deny Karim Benzema.
There were two more Real Madrid opportunities before the half-time whistle, with Dani Ceballos firing wide of the post before Vinicius Junior was denied by the excellent Remiro.
Real Sociedad's goalkeeper was also called into action early in the second half to deny Ceballos, before he blocked another shot from Vinicius, as the hosts piled on the pressure.
The visitors had a chance of their own on the counter-attack in the 51st minute, with Takefusa Kubo driving forward before feeding Sorloth, but the forward dragged his effort wide of the post.
Real Madrid's goalkeeper Thibaut Courtois then did brilliantly to keep out Kubo, before Eder Militao headed into the arms of Remiro down the other end of the field.
Remiro then made a stunning save to deny Vinicius in the 71st minute, with the Brazilian's effort clawed away, and the two teams were still level heading into the final 10 minutes.
Luka Modric was next to come close, dragging an effort wide of the post in the 82nd minute, but the best chance in the final stages came for Real Sociedad, with Courtois keeping out Kubo's effort, as the points were shared at the end of a pulsating contest.
Next up for Real Madrid is a home game against Valencia on Thursday, while Real Sociedad will be back in league action at home to Real Valladolid next Sunday.