The two nations are both outside the top 60 in the FIFA rankings and had surprising runs to the last four, but couldn't overcome the hosts and Argentina respectively to reach the final.
Peru fought valiantly after being reduced to 10 men in the 20th minute when Carlos Zambrano received his second booking and only went down to a long-range wonderstrike from Vargas.
It was Peru's second defeat of the tournament after they were beaten in injury time by Brazil in their opening Group C game.
It wasn't the best start to the tournament, but Gareca's side quickly restored their hopes of reaching the knockout stage with a narrow win over Venezuela before holding Colombia to a goalless draw in the final group match.
Two-time champions, La Blanquirroja might not be as glamorous as many of the other countries on show at the Copa, but they have consistently performed well in recent years, reaching the quarter-finals or better in the last six tournaments.
Recent Copa America form: LWDWL
Recent form (all competitions): DLWDWL
Despite starting the tournament as the second-lowest ranked CONMEBOL side - only Bolivia are four places below them in 89th - Paraguay fell one game short of reaching their second straight Copa final last night.
It was the second time Los Guaranies had met Argentina and they couldn't match the last-gasp 2-2 draw they managed in their opening Group B encounter.
Like their opponents on Friday, La Albirroja have won the Copa twice and also have a strong recent record in the South American competition.
They have only failed to reach the quarter-finals once in the last eight tournaments, but this year was just the second time in that streak that they progressed past the last eight - their 2011 3-0 final defeat to Uruguay in Buenos Aires being the last.
Recent Copa America form: DWDWL
Recent form (all competitions): DDWDWL
Both teams possess two of the best goalscorers of the tournament, with only Vargas's four goals ahead of both Guerrero and Lucas Barrios.
Guerrero got all of his in one game with the quarter-final hat-trick against Bolivia, while Barrios has found the net against the toughest opponents with goals against Argentina and Uruguay.
Barrios has only started one game all tournament for Paraguay and is likely to get the nod here given his impressive scoring record in the tournament.
Peru will be without Zambrano after his red card against the Chileans, with Christian Ramos the man likely to replace him.
Peru possible starting lineup:
Gallese, Advincula, Ramos, Ascues, Vargas, Ballon, Lobaton, Cueva, Carrillo, Farfan, Guerrero
Paraguay possible starting lineup:
Villar, B Valdez, Da Silva, Aguilar, Piris, Gonzalez, Caceres, Ortiz, Benitez, Barrios, N H Valdez
Head To Head
The home side have won the last four meetings between the two in a pair of friendlies and in qualifying for the 2014 World Cup.
Their last Copa meeting came 14 years ago in Colombia when the two played out a back-and-forth 3-3 draw in the group stage.
We say: Peru 2-1 Paraguay
Both sides will want to go out on a high after falling just short of a surprise appearance in the final. We're just siding with Peru because of their dogged recent Copa record and fact that they were much more competitive in the semis.