Argentine champions Boca Juniors snatched a late 1-0 win away to Venezuelan champions Deportivo Tachira while titleholders LDU lost 2-0 at Sport Recife in other games played on Wednesday night.
Cruzeiro's win in Bolivia left them top of Group Five with seven points from three games.
Thiago Ribeiro headed the only goal after 38 minutes and the Foxes survived the last 23 minutes with 10 men after striker Kleber was sent off for kicking an opponent in the head.
Kleber was returning from suspension after being sent off in the 3-0 win over Estudiantes while Cruzeiro also had two men dismissed in last week's draw 1-1 away to Deportivo Quito.
Universitario de Sucre prop up the group with one point from three games.
Substitute Luciano Figueroa headed an 87th minute winner to give Boca Juniors three points from their marathon journey to San Cristobal in the Venezuelan Andes.
The six-times winners, who faced two flights followed by a two-hour bus journey over mountain roads, have six points from two games in Group Two.
Tachira's best chance came when Alejandro Delorte headed against the crossbar in the first half.
Brazil's Sport Recife top Group One with six points from two games after Daniel Paulista and a Paulo Baier penalty gave them the points against LDU.
LDU goalkeeper Jose Cevallos was sent off in the incident which led to the penalty. The Ecuadoreans have three points from two games.
Universidad de Chile handed Bolivian champions Aurora their second straight 3-0 defeat in Group Seven, with Manuel Iturra, Manuel Villalobos and Jose Rojas sharing the goals.
The Chileans lead the group with four points from two games.
