Mikel Oyarzabal and Sheraldo Becker scored for the visitors on Tuesday night in a rematch of the teams' Spanish league game on Saturday, when Sociedad won 1-0 in Vigo.
"We've played against a great team and a great coach, so what we've achieved in these two matches has a lot of value," Sociedad coach Imanol Alguacil said.
Luca de la Torre scored for the hosts with a low shot from inside the box in the 92nd minute.
Celta, coached by Rafa Benítez, were looking to return to the semi-finals for the first time since 2017.
Sociedad, who beat Basque Country rivals Athletic Bilbao in the 2019-20 final, had made it to the last eight the last two seasons but failed to advance.
Oyarzabal opened the scoring with a header from close range two minutes into the match at the Balaidos Stadium.
Becker added to the lead in a fast breakaway in the 66th, making a run from behind the midfield line and finding the net from outside the area as Celta goalkeeper Iván Villar charged from the net.
Sociedad sit in sixth place in the Spanish league, six points from fourth-place Atletico Madrid in the final Champions League place. Celta are 16th, two points from the relegation zone.
Sociedad's win against Celta on Saturday ended the team's four-game winless streak in the league.
On Wednesday (Thursday AEDT), Athletic Bilbao host Barcelona looking to make it to the Copa semi-finals for the fifth straight season. The match pits together the two most successful clubs in the competition. Barcelona have 31 titles and Athletic 23.
Girona, the surprise Spanish league leaders, visit Mallorca in another quarter-final.
Atletico Madrid, who eliminated defending champions Real Madrid in the round of 16, host struggling Sevilla, who sit just outside the relegation zone in the league.
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