Brazil national team goalkeeper Fabio shone for the visitors with five brilliant saves, including two from Estudiantes danger man Juan Sebastian Veron in the first half, before Cruzeiro almost snatched a late winner.
Despite the disappointing outcome, the Argentine side did manage to complete all their home games without conceding a goal and will still fancy their chances of causing an upset in the second leg in Belo Horizonte next Wednesday.
It was a tactical victory for the Brazilians who, packing their midfield, cut the ball supply to Estudiantes' forwards and reduced the Argentine side to counter-attacks and set pieces to try and force a breakthrough.
Three minutes after the interval, Fabio saved a shot from close in on the left from Mauro Boselli, the competition's joint top scorer with seven goals.
"We had the best chances... but their goalkeeper had his night," Veron, whose side are looking for a fourth victory in the competition for Estudiantes, told Fox Sports.
Man of the match Fabio, Cruzeiro's captain, said: "It's a good result, the team played well, it was very difficult. We even had chances to score.
"My hardest save was in the first half face to face (with Perez)."
