Rodrygo would assist his compatriot Vinicius Junior to score the second goal in the 28th minute.
Oleksandr Zubkov converted a ball by Bohdan Mykhaylichenko to pull one back for Shakhtar Donetsk in the 39th minute.
Real Madrid went to the break with the lead and in the second half held on to their advantage to seal three points against a stubborn Shakhtar Donetsk side.
Reactions to Real Madrid
Real Madrid the defending champions now have nine points from three games ahead of the second leg against Shakhtar Donetsk in Ukraine.
Real Madrid star Rodrygo was voted the Champions League Man of the Match after contributing a goal and assist.
Speaking after he received the award, Rodrygo said, "Hello #Madridistas, another win in the Champions League. Thanks so much for your support. Hala Madrid!"
Legendary Brazilian midfielder Kaka hailed the impact of Rodrygo and Vinicius on the team.
He said, “Vinicius & Rodrygo are playing some incredible football right now, they work well with Benzema.
“If Tite has any doubts for calling up Rodrygo for the WC, well, now he shouldn’t. It’s proud for us Brazilians to see Vinícius & Rodrygo perform like this for a club like Real Madrid.”
Rodrygo now has 11 goals in 30 Champions League matches and will aim to add to his tally next week in Ukraine.