Cheryshev, who scored after three minutes, was taken off as soon as Real realised he was ineligible but he faces a ban after being shown a third yellow card playing for Villarreal in the competition last season and failing to serve his suspension.
"We found out at halftime and took him off then as we wanted to do the right thing," Real coach Rafa Benitez told a news conference.
"The club didn't know anything beforehand. We weren't told by the federation or Villarreal."
Cadiz president Manuel Vizcaino said his club would report Real to the Spanish federation.
"The circumstances require a quick decision and the Cadiz board has decided to report Madrid for fielding an ineligible player," he told reporters.
"We have full respect for Madrid but we have taken this decision because we feel it is right for the fans."
Real director Emilio Butragueno said the club's fate now lay with a judge.
"Nobody at the club was informed by the federation, we received nothing from Villarreal and Cheryshev didn't know about the ban," he told reporters.
"We will look into the details fully but I repeat we had no knowledge that he could be suspended. We will study the situation with our legal team and in the end see what the judge decides."
The potential disqualification heaps more pressure on Real Madrid who lost 4-0 at home to arch-rivals Barca in La Liga last month and are six points behind them in the standings after 13 games.
(Editing by Ed Osmond)
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