Rodrygo swiftly went past the Cadiz defenders before hitting the top corner with goals in the 14th and 64th minutes before providing the assist in Bellingham's 74th-minute goal on Sunday.
Madrid moved a point ahead of Girona, who host Athletic Bilbao on Monday.
It was Bellingham's 14th goal with Madrid in all competitions. He leads the league with 11.
Rodrygo was set to start the match on the bench but made the starting 11 at the last minute after Brahim Díaz fell ill with stomach problems.
It was Rodrygo's first game since he said last week that he was the target of racist abuse on social media following Brazil's 1-0 loss to Argentina in South American World Cup qualifying.
Also out for Carlo Ancelotti's team because of injuries were Vinícius Júnior and Eduardo Camavinga, who got hurt during the international break.
It was the second straight league win for Madrid, who are unbeaten in eight matches in the competition.
The result extended Cadiz' winless streak in the league to nine matches, with six losses and three draws. They stayed in 16th place, two points from the relegation zone.
Defending champions Barcelona remained fourth after a 1-1 draw at Rayo Vallecano on Saturday.
Sevilla's winless streak in the league reached seven matches after a 2-1 loss at Real Sociedad, when veteran defenders Sergio Ramos and Jesús Navas were sent off late in the game.
Ramos was shown a straight red card after a dangerous sliding tackle, while Navas was sent off - also with a straight red - for complaining. Ramos was initially shown a yellow card but video review reversed the referee's decision and he ended up being awarded the red.
Sociedad moved to fifth place with the home win. The Basque Country club took the lead with an own-goal by Sevilla goalkeeper Marko Dmitrovic three minutes into the match and added to the lead with Umar Sadiq finding the net with a shot from outside the box in the 22nd.
Sevilla, who dropped to 15th, got on the board with a header by Youssef En-Nesyri in the 60th.
Villarreal ended a two-match losing streak in the league with a 3-1 win against Osasuna, with José Luis Morales scoring a second-half hat trick.
The win moved the team coached by Marcelino García Toral to 12th place. Osasuna, who scored late through Alejandro Catena, dropped to 14th.
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