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.
Niggling fouls on both sides took the flow out of the game, with Cruzeiro penalised 28 times and Estudiantes 23. The home side also failed to take advantage of 10 corners.
Fabio tipped a Veron free kick over the bar in the 11th minute, saved at point blank range from Enzo Perez in the 17th and caught a Veron volley heading for his top left corner just before halftime.
SECOND-HALF SAVES
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.
A minute later, at a corner, defender Leandro Desabato rose perfectly to direct a header towards the near post only for Fabio to dive to his right to turn the ball away.
"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)."
Cruzeiro pushed Estudiantes back into their own half in the second 45 minutes and created two gilt-edged chances late on, both from left crosses, but missed them.
In the 80th minute, Estudiantes keeper Mariano Andujar dived to cut out a low cross only for the ball to fall to Kleber who shot wide of the post.
Three minutes later, a higher cross went over the defence and was headed back into the middle where Wellington blasted it over the bar with the goal at his mercy.
