Universidad de Chile goalkeeper Nestor Conde had appeared to hand Catolica the Group Eight match when he let an easy shot from midfielder Francisco Silva slip under his body and roll slowly into the net in the 70th minute.
"The U" went ahead in the 19th minute when Uruguayan striker Juan Manuel Olivera scored his second goal of the competition but Catolica equalised within two minutes through Matias Rubio.
In the second minute of added time, Argentine playmaker Walter Montillo took a corner from the right to the near post that found Puch who headed the ball sharply past keeper Paulo Garces's into the top corner.
The game in the northern city of Coquimbo was the first football match held in Chile since the massive earthquake that hit the centre and south of the country 10 days ago leaving hundreds of people dead and millions of dollars of damage.
ESTUDIANTES DRAW
Holders Estudiantes were relieved to leave the altitude of La Paz with a point after a 0-0 draw with Bolivar.
The Argentine side had been crushed 4-1 by Alianza Lima in their last away match in Peru in Group Three and could ill afford another morale sapping defeat.
Bolivar fielded half a dozen internationals who had helped Bolivia humiliate Diego Maradona's Argentina 6-1 in April in a World Cup qualifier at the same Hernando Siles stadium in La Paz, 3,650 metres above sea level.
Chances were sparse in the thin air and the closest either side came to scoring was when Bolivia's Brazilian midfielder Charles hit the bar.
Uruguay's Racing overcame the loss of two players to hepatitis to beat Cerro Porteno of Paraguay 2-1 in Montevideo in Group One.
Midfielder Mathias Mirabajes and striker Martin Cauteruccio scored for Racing before Luis Caceres hit a late reply for Cerro after their midfielder Javier Villarreal had been sent off in the 78th minute.
Ecuador's Deportivo Cuenca will host Morelia of Mexico later in a Group Six match.
