Diego Tardelli's goal on the stroke of half-time gave Mineiro a 1-0 win in the second leg at the Estadio Governador Magalhaes Pinto.
Mineiro midfielder Leandro Donizete was sent off five minutes from the end in Belo Horizonte but Levir Culpi's team held on for a 3-0 aggregate success.
Cruzeiro clinched back-to-back league titles with two games to spare after accounting for Goias on Sunday.
But the home side were up against it from the outset in the cup decider as Mineiro attempted to add to their two-goal advantage.
And after dominating the early exchanges, Mineiro put the result beyond doubt via Tardelli in the second minute of first-half stoppage time.
The Brazil international striker was unmarked at the back post and glanced Jesus Datolo's cross past Cruzeiro goalkeeper Fabio.