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.
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.
Ecuador's Deportivo Cuenca will host Morelia of Mexico later in a Group Six match.
