Gareth Bale then responded to his critics by putting the seal on the victory, ending a run of back-to-back draws for Zinedine Zidane's side and stopping Sociedad's 100% start to the season in the process.
The reigning Spanish and European champions were without a host of first-team players for the difficult trip to in-form Sociedad, with Cristiano Ronaldo, Marcelo, Karim Benzema and Toni Kroos all missing from the starting XI.
Sociedad had made their best start to a league season since 1981 with three wins from three heading into the match, but the returning Marco Asensio almost put the hosts behind after 17 minutes when his shot from inside the box was blocked.
Madrid did not have to wait too much longer for the opening goal, though, with Borja capitalising on a rare chance in the first team to drill home his first goal for the club following good work from Sergio Ramos in the Sociedad area.
The goal saw Madrid score for the 73rd match in succession across all competitions, equalling the world record set by Pele's Santos in 1963.
The lead lasted less than 10 minutes before the home side levelled things up, though, as Keylor Navas was unable to hold on to Rodrigues's first-time volley, allowing the ball to slip through his grasp and creep over the line.
Ramos was again one of Madrid's most dangerous attacking weapons and the skipper almost added another goal to his growing tally with 10 minutes remaining of the first half when his header drew a save from Geronimo Rulli.
Sociedad were inches away from taking the lead moments later, though, as Rodrigues again snuck in unchallenged down the left channel but this time saw his effort strike the crossbar with Navas beaten.
It proved to be a pivotal moment in the match as Madrid went up the other end and regained their lead just 23 seconds later when Borja beat two defenders before seeing his low cross deflected past his own keeper by Rodrigues, who had sprinted the length of the pitch to try to help his defence out after hitting the woodwork.
Sociedad's goalscoring form this season left Madrid knowing that another goal may be needed to truly be in control of the contest, but Rulli prevented the visitors from getting it before half time with a good low stop to deny Luka Modric.
The home goalkeeper was called into action again early in the second half when he denied Borja his second of the evening before the Madrid youngster put the rebound wide of the target.
Madrid's third goal did eventually arrive shortly after the half-hour mark, though, as Bale raced clear of the defence before coolly lifting the ball over the onrushing Rulli and into the net.
Lucas Vazquez threatened a fourth 15 minutes from time before Asensio did have the ball in the back of the net, but the goal was ruled out for a block by Bale during the buildup.
Los Blancos were able to settle for a two-goal winning margin in the end, though, surviving some late pressure from the hosts to continue their own unbeaten start to the campaign with a result which lifts them up to fifth in the table.
The result also sees Madrid set a new club record of 11 away league wins in succession, during which time they have scored 38 goals.