As had been expected, it was Argentina who started the encounter in Chile strongly, with Paraguay having few answers to a quality Argentinean attack.
Manchester City striker Sergio Aguero opened the scoring in the 29th minute when he showed off his goal-poaching instincts by intercepting a poor backpass from Paraguay defender Miguel Samudio before slotting home.
Seven minutes later Lionel Messi made it two from the penalty spot after Angel Di Maria had been brought down in the box.
It appeared that Argentina were cruising, but Paraguay lunched a stunning comeback.
Nelson Valdez began the fightback when he pulled one back in the 60th minute when his long-range effort beat Sergio Romero in the Argentina goal.
Sensing that they had the opportunity to salvage something from the contest, Paraguay attacked strongly and were rewarded for their efforts in the 90th minute when Lucas Barrios got on the end of a knock-down from Paulo Da Silva before finding the back of the net with a low shot.
Defending champions Uruguay got their Copa America defence off to a winning start with a 1-0 win over Jamaica in Antofagasta.
Uruguay were not at their best on the day, but Cristian Rodriguez's 52nd-minute effort from close range was enough to secure the three points.
After a first half that contained few real chances, Rodriquez was on hand to get on the end of Jose Maria Gimenez's header from a free-kick and made no mistake with his shot.
Jamaica had chances to level the scores, with Deshorn Brown and Kemar Lawrence both coming close. However, it was Uruguay who should have added to the scoring, with Christian Stuani somehow conspiring to miss an open net in the 88th minute.