The team coached by Ricardo Gareca controlled the match but wasted chances until the 89th minute, when Mauro Obolo teed up forward Juan Manuel Martinez to volley in the only goal.
Gareca said: "We gave a great performance. I'm proud of this team because the players always searched for the victory until the last minute and, thank God, they found it in the end."
Matchwinner Martinez added: "These three points are very important, but we have a lot of work still to do in the group."
Velez have nine points, five more than Deportivo Quito and Mexican giants Guadalajara and six more than Uruguay's Defensor Sporting.
A superb performance from Neymar helped Brazilian side Santos defeat Peruvians Juan Aurich 2-0 at home to go clear at the top of Group One.
The coveted striker scored one goal and captain Edu Dracena hit another to give the team coached by Muricy Ramalho a comfortable win despite heavy rain at the Pacaembu stadium.
Neymar later complained about the physicality of the opponents, saying: "They were very violent, the referee must stop these kind of attitudes for the good of the game."
Santos moved to nine points, two better off than fellow Brazilians Internacional and Bolivians The Strongest. Aurich have no points and cannot advance from the group.
Colombians Atletico Nacional drew 2-2 at home against Argentinians Godoy Cruz in their Group Eight match in Medellin.
Luis Fernando Mosquera took advantage of a defensive mistake to give Nacional the lead after two minutes, but Jorge Curbelo and Facundo Castillon put the visitors 2-1 up.
Mosquera's penalty in the 61st minute levelled the match, while Godoy Cruz ended the match with 10 players as defender Leonardo Sigali was sent off in the 90th minute.
Atletico Nacional head Group Eight with eight points, with Godoy Cruz second on five points, ahead of Universidad de Chile and Penarol.