Carlos Augusto Lobaton fired a powerful shot from just inside the penalty area in the 101st minute and fellow midfielder Juan Manuel Vargas sealed the victory with a shot from near the penalty spot in the 111th.
Peru will next face either Argentina or Uruguay, who met late Saturday in Santa Fe.
Colombia, a title favorite after playing well in the first round, hadn’t allowed a goal after its first three group matches.
Lobaton entered the match at halftime and scored after Colombia goalkeeper Luis Enrique Martinez failed to hold on to a long ball crossed into the area. Lobaton’s right-footer struck the crossbar before going into the net.
Vargas closed the scoring in a breakaway after another mistake by Martinez, who gave the ball away with a bad pass that left Peru with a two-on-one situation for an easy goal.
Falcao missed the penalty which was called after defender Alberto Rodriguez fouled striker Dayro Moreno inside the area, dragging him down by the arm as he tried to reach for a long ball.
Colombia had most of the dangerous opportunities throughout the match at Estadio Mario Alberto Kempes.
Moreno had one of the best chances just two minutes after the missed penalty, firing a powerful right-footer from inside the box into the left post of Peru goalkeeper Fernandez Valverde.
Colombia struck the woodwork again two minutes into second-half stoppage time when Fredy Alejandro Guarin cleared two defenders on the left side before entering the area and striking the crossbar with a hard shot.
Peru threatened with long-range shots throughout regulation, including two by Vargas in the first half and then two more by William Chiroque and Adan Balbin in the second.
Colombia, who last won the title in 2001 at home, reached the quarterfinals by winning Group A ahead of host Argentina. Peru, which hasn’t won the tournament since 1975, was only third in Group B, behind Chile and Uruguay.
Peru advanced despite playing without top strikers Jefferson Farfan and Claudio Pizarro, who didn’t make it into the tournament because of injuries.