Oleksandr Zubkov gave the Ukrainian visitors hope with a well-taken header.

However, Shakhtar could not find a leveller as the Spanish giants moved closer to securing their spot in the knockout rounds.

Real Madrid made the perfect start when Rodrygo struck from the edge of the box after being found by the impressive Aurelien Tchouameni.

Karim Benzema, Federico Valverde and Rodrygo linked up well for the latter to tee up Vinicius Jr for the home side's second.

Despite their first-half dominance, Carlo Ancelotti's side went into the break only one up after Zubkov headed Bogdan Mykhaylychenko's cross past Madrid's Ukrainian keeper Andriy Lunin, but the visitors were unable to salvage a point.

Shakhtar remain second in Group F, one point above RB Leipzig and face Madrid in Warsaw in the return fixture on Tuesday.