Coach Alejandro Sabella, whose side had already booked their place in Brazil next year, was without the likes of Lionel Messi, Gonzalo Higuain,
Fernando Gago and Javier Mascherano, and it appeared to be hurting them as Claudio Pizarro gave Peru the lead.
But Lavezzi netted twice before the break to turn things around and Rodrigo Palacio added a third at the start of the second half.
Peru coach Sergio Markarian fielded several debutants with an eye on the future but it was veteran striker Pizarro who opened the scoring after 20 minutes, beating Sergio Romero with an outstanding effort from outside the box.
The lead lasted just two minutes though as Lavezzi took advantage to sink the rebound when Diego Penny blocked a goalward header from Argentina defender Federico Fernandez.
And 10 minutes before the break the turnaround was complete when Palacio advanced down the left and found Paris St Germain striker Lavezzi for an easy finish.
Palacio connected with Marco Rojo's cross immediately after the restart to net Argentina's third, but despite struggling to contain the attacking talents of Palacio, Lavezzi and Sergio Aguero, the Peru defence did not concede again.
"I am very happy that I could take advantage of the boss' support," said Lavezzi. "I think we are doing well and that gave us confidence for the future."
Argentina visit Uruguay on Tuesday for their final qualifier, while Peru face Bolivia.