Estudiantes, South American champions in 1968, 1969, 1970 and 2009, suffered their previous worst defeats, 4-1 away to Argentina's Independiente in 1984 and Alianza Lima of Peru last year.
Striker Wallyson benefitted from a deflection off defender German Re with the ball looping over goalkeeper Agustin Orion into the net after 48 seconds to put twice champions Cruzeiro ahead at home in the Aligator's Arena ground.
"We had to run faster and work harder than them, who have dangermen such as (winger) Enzo Perez, who is in good form, (striker Gaston) "Gata" Fernandez and (Juan Sebastian) Veron," Montillo said.
"We didn't expect this result. But this is only the first three points of 18 in play (in the group)," added the man of the match.
Midfielder Roger scored Cruzeiro's second, laid on by Montillo, in the 17th minute then returned the compliment for the Argentine in the 38th.
Montillo scored again the 58th with a low shot and Wallyson completed Estudiantes' humiliation six minutes from time when a poor clearance went in off his chest.
Brazilian holders Internacional were held 1-1 away by Ecuador's Emelec in a Group Six match at the George Capwell stadium in Guayaquil.
Defender Fernando Gimenez headed an equaliser for Emelec in the fourth minute of added time after Inter's Argentina midfielder Mario Bolatti, signed from Fiorentina a week ago, had given them the lead in the 79th with a header from a corner.
