Bolivian attacker Marcelo Moreno bagged a brace as the Brazilian Serie A champions produced a masterclass in the first leg of their last-16 clash at the Estadio Governador Magalhaes Pinto.
Cruzeiro - seven points clear in Brazil's top flight - went into Wednesday's fixture against Campeonato Alagoana opponents Santa Rita in the midst of a 10-game unbeaten streak in all competitions.
And Marcelo Oliveira's men extended that run thanks to a dominant first half that yielded three goals.
Moreno opened the scoring after eight minutes in Belo Horizonte, looping a header over goalkeeper Jeferson from Alisson's cross.
Centre-back Dede doubled Cruzeiro's lead in the 18th minute, when he headed home from a corner.
Former Arsenal and Real Madrid man Julio Baptista made it three with a header across goal in the 35th minute before Moreno doubled his tally nine minutes after the interval.
Henrique capped the lopsided win with three minutes remaining, firing a half-volley past Jeferson.
It was a different story for Serie A rivals Corinthians, who must overturn a one-goal deficit in the return leg after losing 1-0 at Bragantino on Wednesday.
Sandro Mendonca's 56th-minute goal gave the second-tier side a surprise 1-0 win in Cuiaba, while it condemned Corinthians to back-to-back defeats.
Corinthians' woes were compounded in the third minute of injury time, when midfielder Ferrugem received his marching orders.
Atletico Mineiro, Santos, Coritiba, America RN and Ceara were also among the first-leg winners midweek.
Mineiro were 1-0 victors at Palmeiras on Wednesday, with Luan's glancing header in the 71st minute the difference after Henrique missed a penalty for the home side in the first half.
David Braz and Milan loanee Robinho were both on target in the opening half as Santos upstaged Luiz Felipe Scolari's Gremio 2-0 away from home on Thursday.
Coritiba will take a three-goal advantage into the second leg after accounting for visitors Flamengo 3-0 on Wednesday.
The defeat ended Flamengo's four-game winning streak.
Serie B's America RN were 3-0 winners at home to top-flight outfit Atletico PR on Wednesday.
Rodrigo Pimpao, Max and Thiago Cristian found the back of the net for America RN.
Ceara netted two crucial away goals as they prevailed 2-1 at Botafogo on the same day.
Meanwhile, hosts Vasco da Gama were held to a 1-1 draw by ABC on Tuesday.
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