Colombia moved to second in the CONMEBOL group standings, three points behind Argentina who drew 1-1 in Ecuador despite having Javier Mascherano red carded four minutes from time.
Sergio Aguero gave Argentina the lead from the penalty spot early on before Segundo Castillo headed the home side level.
Colombia now lie seven points above fifth-placed Uruguay in the battle to secure a place in Brazil next year.
Radamel Falcao opened the scoring for Colombia with a 12th-minute penalty after Yoshimar Yotum fouled Camilo Zuniga in the box.
Teofilo Gutierrez secured the win one minute before half-time.
Peru ended the match with 10 men as defender Carlos Zambrano was sent off after 66 minutes.
Monaco striker Falcao, who has netted seven goals in the current World Cup Qualifying campaign, told reporters in Barranquilla: "To be the tournament's top scorer comes second. What it matters is that Colombia reach the World Cup.
"We managed to be patient against a team that planned the game very well."
Chile moved level with Ecuador on 21 points by recording their seventh win in the competition with a 3-1 home triumph over Bolivia.
Jorge Sampaoli's Chile, whose record shows no draws in the tournament from 13 games, took the lead with Eduardo Vargas' 15th-minute strike.
Two minutes later, Alexis Sanchez doubled the lead with a header but Marcelo Moreno Martins reduced the deficit after 31 minutes.
In second half injury time, Jean Beausejour crossed the ball and Arturo Vidal headed home to make the final score 3-1. The game saw the return of David Pizarro to the Chilean national team after an eight-year absence.
In Puerto Ordaz, Uruguay claimed a crucial win by defeating hosts Venezuela 1-0. A 27th-minute goal from Edinson Cavani secured a first away win for the Charruas in the campaign.
Only goal difference sees Uruguay above Venezuela, as both sides are level on 16 points. Venezuela midfielder Tomas Rincon was sent off three minutes before the final whistle.
Fifth place will earn a knockout tie with a CONCACAF side for a place in the finals.