Angel Di Maria took over as captain in Messi's absence and shone, providing two assists before halftime
Traveller Balram Menon shares his experiences of travelling in Bolivia, Brazil, Peru, Italy, France and other countries in this series.
The result means that Argentina will face Ecuador in the last eight on Saturday while Uruguay, who beat Paraguay 1-0, will play Colombia the same day.
Roberto Firmino scored twice and Marquinhos and Philippe Coutinho also got goals as Brazil routed Bolivia 5-0 in their opening World Cup qualifier.
Peru were the superior team in the second half and could have scored more but for a series of excellent saves by Bolivian goalkeeper Carlos Lampe.
Philippe Coutinho scored a second half double and Everton added a cracking third as Brazil recovered from a sleepy start to beat Bolivia 3-0 in their Copa America opener.