The news before kickoff was that Jude Bellingham and Aurelien Tchouameni were not risked despite being close to a return to action.
Unfortunately for Los Blancos, another name was added to their absentee list in the opening half hour when Brahim Diaz was forced off due to a muscle injury.
Real Sociedad went closest to breaking the deadlock, with Luka Sucic rattling the woodwork before Sheraldo Becker struck Thibaut Courtois's crossbar.
La Real experienced more frustration in the opening 30 seconds of the second period when Sucic saw his left-foot strike bounce off the far post.
The hosts were ultimately made to rue those missed chances as Real Madrid stamped their authority on the contest with two successful spot-kicks.
First, Vinicius dispatched his 58th-minute spot-kick to break the deadlock, before Mbappe kept his composure to convert his penalty and wrap up all three points for Carlo Ancelotti's side.
The win extends Real Madrid's unbeaten La Liga run to 37 matches and moves them to within one point of leaders Barcelona, who will face Girona on Sunday.
Meanwhile, Real Sociedad have lost each of their three home games this season and are left languishing in 16th position.
REAL SOCIEDAD VS. REAL MADRID HIGHLIGHTS
Vinicius goal vs. Real Sociedad (58th min, Real Sociedad 0-1 Real Madrid )
Real Sociedad have hit the woodwork on three occasions, but they find themselves behind following a Vinicius penalty. Gomez is penalised for handling Arda Guler's powerful shot, giving Vinicius the chance to find the net from the penalty spot despite Remiro's best efforts.
Mbappe goal vs. Real Sociedad (75th min, Real Sociedad 0-2 Real Madrid)
Real Madrid double their advantage from another penalty. The official heads over to the monitor and rule that Jon Aramburu's challenge on Vinicius is worthy of a spot-kick. On this occasion, Vinicius hands penalty duties over to Mbappe, who sends Remiro the wrong to score his third La Liga goal.
MAN OF THE MATCH - VINICIUS JUNIOR
Vinicius showed great composure to convert his penalty to register his second goal of the La Liga season after previously scoring against Las Palmas.
The attacker then played a key part in the second goal, with his skill and nimble footwork drawing a foul from Aramburu in the Real Sociedad box.
As well as winning the second penalty, the Brazilian won more duels than any other player on the pitch and offered a constant attacking threat throughout the contest.
REAL SOCIEDAD VS. REAL MADRID MATCH STATS
Possession: Real Sociedad 48%-52% Real Madrid
Shots: Real Sociedad 11-16 Real Madrid
Shots on target: Real Sociedad 2-6 Real Madrid
Corners: Real Sociedad 6-6 Real Madrid
Fouls: Real Sociedad 12-9 Real Madrid
BEST STATS
WHAT NEXT?
After claiming their first away win of the season, Real Madrid will return to the Santiago Bernabeu for Tuesday's Champions League meeting with Stuttgart.
As for Real Sociedad, they will look to produce a positive response when they face Mallorca at San Moix in Tuesday's La Liga clash.