The two nations, meeting for the first time, won their opening games against Greece and Japan respectively and Colombia's success in an entertaining contest puts them on six points at the top of the group, with Ivory Coast on three. (Agencies)
The match only really came to life in the second half and Rodriguez broke the deadlock in the 64th minute when he beat substitute Didier Drogba to the ball at a corner and sent it arrowing into the net off goalkeeper Boubacar Barry.
Rodriguez dispossessed Serey Die six minutes later and Teofilo Gutierrez sent Quintero clear to finish coolly past Barry and send the thousands of yellow-clad Colombia supporters thronging the national stadium in Brasilia into ecstasy.
Gervinho, who also scored against Japan, pulled one back for the Africans in the 73rd minute but Colombia held firm to put one foot firmly in the last 16.
After a cagey opening few minutes in the giant arena in the Brazilian capital, Colombia started to push forward, with midfielder Juan Cuadrado at the heart of their attacks.
It was Cuadrado's ball out to the left that released Rodriguez in the 28th minute but his fine cross was scuffed wide by striker Gutierrez unmarked in front of goal.
Ivory Coast had their first genuine chance three minutes later when lively fullback Serge Aurier cut inside his marker and his low shot forced a good save from Colombia goalkeeper David Ospina.
Ivory Coast talisman Drogba, who was introduced on the hour, was at fault for the opening goal and made little impact during the rest of the game.
After Colombia's quickfire double and Gervinho's response, Ivory Coast pressed hard for an equaliser, Salomon Kalou, another substitute, shooting straight at Ospina when well placed five minutes from time.
The two nations, meeting for the first time, won their opening games against Greece and Japan respectively and Colombia's success in an entertaining contest puts them on six points at the top of the group, with Ivory Coast on three.