Ivory Coast were unable to find an equaliser after Gervinho had pulled a goal back and Jose Pekerman’s side's qualification for the last 16 of the World Cup was later confirmed after Japan failed to beat Greece.
Teofilo Gutierrez missed the best chance of an open first half for Colombia before Juan Cuadrado’s shot was tipped onto the bar by Boubacar Barry on 59 minutes.
Colombia, who beat Greece in their opening game, took the lead on 64 minutes when Cuadrado curled in a corner and Rodriguez powered a header past Barry.
Just six minutes later, Colombia increased their lead when Gutierrez slipped in Quintero to finish past Barry for his first international goal after Rodriguez had pinched possession.
Ivory Coast replied on 73 minutes through Gervinho’s fine individual effort, but Colombia held out to stand on the brink of a place in the knockout rounds.
Pekerman named the same starting XI which helped Colombia to defeat Greece 3-0, with captain Mario Yepes earning his 100th cap, while Ivory Coast again left talisman Didier Drogba on the bench.
It was Colombia who made the more positive opening, with Gutierrez firing wide from the edge of the penalty area and Didier Zokora needing to make an important block after Cuadrado had got away down the right.
On 23 minutes, there was the first opening for Ivory Coast when a powerful run from captain Yaya Toure released Gervinho, but he was quickly hustled off the ball.
Colombia should have been ahead on the half-hour when a counter attack down the left saw the ball played into the penalty area for Gutierrez.
However, the River Plate forward, who looked to be offside, made a complete hash of his shot from six yards out and stabbed the ball wide.
Ivory Coast full-back Serge Aurier, linked with a summer move to Arsenal, sent in a low drive from 18 yards, which was comfortably saved by Colombia goalkeeper David Ospina.
At the start of the second half, English referee Howard Webb waved away somewhat hopeful penalty appeals from Ivory Coast after Roma forward Gervinho went down under a challenge from Yepes.
On 55 minutes, Zokora was shown a yellow card for a late challenge on Cuadrado, which meant the one-time Tottenham defender would miss the final group match against Japan.
An ambitious bicycle kick from Swansea forward Wilfried Bony did not come off, before at the other end, Cuadrado's shot was pushed onto the bar by Ivory Coast goalkeeper Barry.
On the hour, Ivory Coast manager Sabri Lamouchi made a change when Drogba replaced Bony.
It was Colombia, though, who took the lead on 64 minutes when Monaco forward Rodriguez headed home from a corner.
There was immediately a second substitution for Ivory Coast as former Leicester and Leeds wideman Max Gradel was replaced by ex-Chelsea forward Salomon Kalou, however, Colombia doubled their lead on 70 minutes with a breakaway goal.
Gutierrez fed Quintero after Rodriguez had closed down some poor Ivory Coast defending on the half-way line, and the Porto forward took the ball on to the edge of the penalty area before drilling it past Barry.
Ivory Coast needed a response - and got one through Gervinho's second goal of the World Cup on 73 minutes.
The winger, much maligned during his spell with Arsenal, collected the ball on the left and weaved into the penalty area, rode three challenges before smashing a shot inside the near post which Ospina got a hand to but could not keep out.
Ivory Coast were lifted by the goal, and went in search of an equaliser, substitute Mathis Bolly just unable to guide a diving header on target before Kalou shot straight at the goalkeeper.
Drogba almost got on the end of a long through ball during stoppage time, but Ospina was out quickly to make a clearance, leaving Ivory Coast likely needing to avoid defeat by Greece in their final group game to join Colombia in the next round.