Father and son will be reunited, albeit on opposite sides of the ball, when Madrid visits the Basque Country club.
Zinedine Zidane's side nearly made the game safe when Cristiano Ronaldo's drive cannoned back off the post, though up the other end, Alaves were also denied by the woodwork, Alfonso hitting the frame of the goal from distance.
Sergio Ramos almost doubled the visitors' lead after Cristiano Ronaldo cleverly chipped a free kick into the box, but the unmarked defender had his header saved.
Alaves looked set to extend Madrid's patchy start to the season when captain Manu Garcia equalised before the break, but Ceballos quickly responded with his second to help secure just a third win in six league matches.More news: Spinner Kuldeep Yadav takes hat-trick against Australia
But Ceballos soon eased the pressure on the visitors with the opener, wriggling out of a challenge with Ibai Gomez in the area to slide a finish inside the left-hand post.
Manuel Garcia's 40th-minute goal temporarily drew Alaves level.
Ceballos caught Madrid's eye in the offseason after he was named the best player at the Under-21 European Championship, where he helped Spain reach the final before losing to Germany 1-0.
Aside from Ramos' efforts, Real's other clear chances fell to Cristiano Ronaldo. Spanish born Ghanaian striker Inaki Williams scored two goals for the latter.
The goalkeeper was again called into action to tip over Nacho Fernandez's aerial effort, before the tempo died down, with Madrid playing keep-ball and giving Alaves the run-around.