The 37-year-old Chilean struck five minutes into the second half at the Ramón Sánchez-Pizjuán Stadium, converting Neal Maupay's lay-off inside the area to give Sevilla a first-half-time lead that would prove decisive.
Sanchez was named man of the match and was direct about the nature of the evening's stakes when speaking to DAZN afterwards.
"I think I've always had to play finals and moments like this in my life. It serves me because I always want to win."
He described the goal in characteristically uncomplicated terms.
"I stepped out of the offside and found the space. I hit it with my heart."
The veteran also acknowledged taking a knock during the match.
"I'm feeling a little dizzy because I took an elbow to the head for being a bit shorter."
Sevilla had entered the game two points from safety, having lost five of their previous six matches. The win moved them to 37 points, one above Alavés in the third and final relegation place, with the chaotic battle in the lower reaches of the table continuing to tighten.
A late goal from Lucien Agoumé looked set to settle any remaining nerves, only for VAR to rule it out for offside. Sociedad threatened a leveller in the final minutes but could not find a way through.
Speaking in Sevilla's official club media, Sanchez reflected on his contribution at this stage of his career.
"There's a reason they brought me here. I have experience. I'm still proving myself, I'm in good physical shape."
He also drew attention to the spirit in the Sevilla squad, pointing specifically to goalkeeper goalkeeper Yassine Bounou's understudy Adriá Suazo.
"The other day, when we lost in the last minute, I saw many players like Suazo crying. He felt it, and you always need a player like him."
Sevilla's remaining fixtures bring Espanyol at home, then trips to Villarreal and Celta Vigo, with Real Madrid visiting the Pizjuán in between.