The stricken Cantabrian club, which has spent the majority of its 101-year history in the top flight, is currently in administration following two consecutive relegations.
Despite an amazing cup run that has included victories over La Liga sides Sevilla and Almeria, the players haven't been paid since September.
The squad warned on Monday that they would refuse to play the match if Lavin and the rest of his board didn't resign.
Both sides took to the field of play and kicked off as normal.
However, after Sociedad had kept possession for 40 seconds without any challenge from the Racing players, referee Gil Manzano called the game off when Santander captain Mario Fernandez informed him his side would take no further part.
Sociedad were leading 3-1 from the first leg last week and are expected to be awarded a 3-0 win on the night to progress to the last four where they will meet Barcelona.
"We have given up what we like most, which is to play," said Racing forward Mariano Sanz.
"We wanted to play but we couldn't let them stand on our rights. To play in the quarters was a dream for us, everything had gone so well to get there.
"It is a shame to end it like this but we were clear in what we were doing."
Far from being angered by the player's stance, the Racing fans gave the squad a standing ovation as they did a lap around the pitch before heading back down to the dressing rooms.
It is not the first time that the players have used their Cup success to protest over their plight as they began the first leg of their tie against Almeria by standing motionless for the first 20 seconds of the match before beginning to play.
The supporters also showed their frustration at the board in the same game by storming the directors' box to demand Lavin's resignation.