The Real Madrid captain scooped the prestigious prize for his successes on the pitch and his work in promoting sports in the Spanish capital over the years.
"He is making great strides", Real coach Santiago Solari said of Vinicius.
"We have to be realistic, it's going to be very hard now", Leganes midfielder Ruben Perez said.
A push on Alvaro Odriozola was adjudged as a foul inside the box by the referee Jesus Gil Manzano, and Sergio Ramos made no mistake from 12 paces out as he sent Pichu Cuellar the wrong way to open the scoring a minute shy of half-time.
Vazquez scored the second in the 68th after a pass by Vinicius Junior, who 11 minutes later sealed the victory with a neat volley after a cross by Odriozola.
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Leganes couldn't use goalkeeper Andriy Lunin and young midfielder Oscar Rodriguez because they are with the club on a loan from Madrid.
Last season, Madrid were stunned 2-1 at home by Leganes and went out on away goals despite having won the first leg 1-0, and Butragueno was keen to guard against another shock in the return fixture next week.
The teams had drawn by the same score in their league match in Girona earlier this season.
Solari, who has faced criticism for the team's drab start to 2019, also said the team felt a sense of release after the victory after the Sociedad defeat and last week's 2-2 draw at Villarreal.
On Thursday, four-time defending champion Barcelona visits Levante.
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