Veron netted the rebound after Alianza goalkeeper George Forsyth saved his penalty to put the Argentine side top of Group Three with 13 points from their six matches.
Alianza had defender Walter Vilchez sent off three minutes into stoppage time for a foul on winger Enzo Perez to give away the penalty.
The Peruvian team, who had crushed Estudiantes 4-1 when the teams met in Peru in February, finished second with 12 points which they hope will be enough to see them through as one of the six best second-placed teams in the eight groups.
Another Peruvian side, Juan Aurich are third with six points after losing 2-0 against bottom team Bolivar in La Paz. The Bolivian side, winning for the first time, finished with four points.
"I feel bitter because we deserved to draw. Football is like that and it's over," goalkeeper Forsyth told reporters.
The Estudiantes players were full of praise for Alianza.
"It cost us a lot to win. We played against a great team who beat us well in Lima," said midfielder Rodrigo Brana.
With Argentina coach Diego Maradona looking on, Estudiantes came up against a well organised and tactically astute Alianza team, whose winger Wilmer Aguirre was a handful on the counter-attack. Aguirre had scored a hat-trick in the teams' first meeting.
