Defender Marcos Rojo and midfielder Javier Pastore gave them a 2-0 lead within the first half hour, both goals set up by an inspirational Lionel Messi.
Paraguay pulled a goal back through Lucas Barrios before the Argentines ran riot in the second half as Di Maria scored twice in six minutes before strikers Sergio Aguero and Gonzalo Higuain added their fifth and sixth.
Argentina will meet Chile in Saturday's final in Santiago while Paraguay will face Peru on Friday in the third-place play-off.