Mineiro were too strong for the four-time champions at the Estadio Raimundo Sampaio, with Luan and Argentinean midfielder Jesus Datolo on target in Belo Horizonte.
Levir Culpi's Mineiro - 10 points adrift of league-leading Cruzeiro in the Brazilian Serie A - went into the match with form on their side.
Mineiro had not lost in six games against Cruzeiro in all competitions prior to kick-off, and the home side wasted little time gaining the ascendancy in the competition decider.
After Cruzeiro failed to clear their lines sufficiently in the ninth minute, Marcos Rocha whipped in a cross from out wide for Luan to head home from inside the six-yard box.
It got better for Mineiro after half-time, as Datolo doubled the lead on the hour-mark.
Carlos laid the ball off for Datolo, who curled his effort underneath Cruzeiro goalkeeper Fabio.
The return leg is at the Estadio Governador Magalhaes Pinto on November 26.
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