The Brazilian winger gave Real the lead in the second minute following a pass from his compatriot Vinicius Junior.
The European champion did not take advantage of the early lead to allow Osasuna, ranked tenth in the Spanish Championship, to equalize in the 58th minute on a powerful shot from Lucas Toro.
But Rodrigo sealed the victory 20 minutes from time by avoiding the marker to score from close range, after a distinguished individual effort from Venice, penetrating from the left.
The title boosts the morale of Real Madrid, who are very close to losing their title in the local league against their Barcelona rival, before welcoming Manchester City next Tuesday in the semi-final first leg of the Champions League.
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