Siboldi's side had secured a 4-0 advantage in last week's first leg of that tilt, only to squander the lead in the return leg on Sunday with a 4-4 aggregate that allowed Pumas to reach the "Guard1anes 2020" finals against Leon.
The loss was seen as the latest example of a "cruzazuleada," a word that has entered the Spanish lexicon at the expense of the team to define a big defeat from an advantageous position.
"After analyzing the most recent events with my family and staff, I have decided to take a step back," Siboldi said on Friday.
Club president Victor Velazquez called the loss the worst in the club's history and that it would take "radical and urgent decisions."
Siboldi added that he was proud of what his side were able to achieve despite not winning a championship.
"My players are professionals and are not mediocre, they gave everything to win. I will not permit that my honor and values are questioned, nor of my staff and of my players. We had great hope to win this tournament," the Uruguayan said.
Cruz Azul is one of Liga MX's traditional "big four" clubs -- along with Pumas, America, and Chivas -- but is without a league title since 1997. Since then, they have finished runners-up on six occasions.
Despite the loss, the club is still in contention for the 2020 CONCACAF Champions League title, which will re-start in Orlando next week following its suspension due to the coronavirus pandemic.