Playing at their Minerao home, the Brazilian club overturned a 2-0 first-leg deficit thanks to goals from Jo and Leonardo Silva - the latter coming with just three minutes of normal time remaining.
And they went on to convert four penalties in the shootout after extra-time while their Paraguayan opposition netted just twice, to seal South America's premier cup competition for the first time in their history.
Cuca has subsequently hailed the resilience and belief shown by his side, citing the numerous obstacles they had to overcome on the way to victory.
"We've managed the first-leg defeat quite well - as we did against Newell's (Old Boys) as well. We worked our emotional side during the week keeping everybody positive and faithful," he said.
"We did everything right. But the match was really tense because we are all human beings. We had the obligation of getting this trophy for the first time. So we couldn't play the final naturally with patience."
The 50-year-old also explained that the club's historic achievement has not yet sunk in.
"Today Atletico is a winning club. No one can say the opposite. Especially after all we've been through in this competition," he added.
"We saved a penalty kick in the 93rd minute (of the quarter-final win over Tijuana), we won in shootouts, we lost 2-0 in two first legs including in the final, so today this changed.
"We have changed a club history. A century (old) club that now is a lucky club. And so am I. We haven't realised this achievement yet. It is too much. We are all numb at this moment. But we have to be thankful."
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