In a match of few chances, Real finally gained the lead when Casemiro's intelligent finish broke the deadlock after 57 minutes.
However, Sevilla deservedly earned an equaliser just seven minutes later through Luuk de Jong's curling shot.
The game remained level for just five minutes as Casemiro completed his brace with a 69th-minute header.
An even first half saw both sides struggle in the final third, but the visitors appeared mobile and motivated to pressurise the hosts.
Real had few opportunities to break the deadlock, but Casemiro's long-range effort caused Sevilla goalkeeper Tomas Vaclik concern midway through the first half.
The away team believed they had taken the lead after De Jong's back-post header beat Thibaut Courtois. However, after the referee reviewed the VAR monitor, the effort was disallowed because Nemanja Gudelj fouled Eder Militao in the buildup.
Both sides only managed one shot on target each before the interval, and Real appeared jaded after their 2020 Spanish Super Cup victory in Saudi Arabia.
Journalist Dermot Corrigan summarised the first-half action:
Los Blancos and Los Nervionenses both took their time to find traction after the restart, but it was the latter who controlled possession. However, Real grabbed the lead after eventually putting together a worthy attack.
Casemiro combined with Luka Jovic on the edge of the box, and the Brazil international expertly lifted the ball over Vaclik as he attempted to save.
Going behind was cruel on Sevilla, but they responded in the best way possible to level the contest.
Real failed to clear their lines on the edge of the box, allowing De Jong to wriggle past defenders and curl his left-footed shot past the helpless Courtois.
The Dutchman claimed a unique landmark with his superb finish, per OptaJohan:
Zinedine Zidane's men took no time to answer, and it was soon 2-1 after excellent work from Lucas Vazquez. The winger's cross was perfectly executed, and Casemiro scored his second with an unmarked header.
It was a moment of poor defending by Sevilla, who were convincingly punished by the hosts.
La Liga highlighted a special moment for Real's matchday hero:
Casemiro nearly completed his first-ever hat-trick, but he dragged his shot wide.
Although Sevilla continued to search for a way forward, they couldn't find a second equaliser.
Real's victory means they are now 17 games unbeaten in all competitions and have gone three points clear of Barcelona at the top of La Liga, but the champions can close the gap on Sunday when they host Granada.
Both teams are back in top-flight action next weekend. Sevilla host Granada on January 25, and Madrid travel to Real Valladolid on the following day.