Nacho Fernandez and Marco Asensio struck to help champions Madrid cut Barcelona's lead at the top of the table to 10 points, before the Catalans visit Getafe on Sunday.
Both sides hit the woodwork in the first half but Madrid overran their opponents, with only Cadiz goalkeeper David Gil keeping them at bay.
Eventually, when it seemed they had tried everything, Nacho broke the deadlock in the 72nd minute with a drive from distance, with Asensio adding another four minutes later.
"Good attitude, good commitment, good combinations, the team played very well," said Madrid coach Carlo Ancelotti.
"It was a game that could have been a trap - the players were concentrating well, and we played well."
Ancelotti rang the changes at the Nuevo Mirandilla to rest players for the visit to Stamford Bridge for the quarter-final second leg, including veteran midfielders Toni Kroos and Luka Modric and winger Vinicius Junior.
Madrid suffered a 3-2 home defeat by Villarreal last weekend before the 2-0 first leg win over Chelsea, and Ancelotti admitted his team were not motivated enough.
On this occasion they arrived with the right mindset and on another day would have ended the first half several goals in front.
Gil made stunning saves to deny Karim Benzema, Rodrygo and Asensio.
Madrid were racking up the shots and, finally, their 26th effort of the night went in.
It came from an unlikely source, in veteran defender Nacho, who drove home from the edge of the box into the bottom corner.
With Cadiz's resistance broken, Madrid quickly added another.
Fede Valverde carried the ball forward and fed Asensio, who finished clinically.
It was a satisfying evening for Ancelotti and his team, leaving them in good shape to continue their European adventure, aiming to finish the job in London and reach the semi-finals.
Despite his other rotations, Ancelotti left Benzema on the pitch for the full 90 minutes, a surprise given he has had fitness issues this season.
"I saw him doing so well, he was so fresh," the coach told reporters.
"I told him I'll start you and when you are tired, I'll take you off, but he was good until the end.
"He was a bit sad not to have scored, but he combined spectacularly with Rodrygo and it was a pleasure to watch him - it was hard for me to take him off because I wanted to enjoy his quality on the pitch."
Defeat leaves Cadiz 15th, four points clear of the drop zone.
