The defender lashed home a stunning free kick after Thiago Silva's early goal, with James Rodriguez scoring a penalty before the Barcelona star was forced off in the dying minutes
Tipped by some to exit against a side who have been the talk of the tournament, Luiz Felipe Scolari's men progressed thanks to goals from Thiago Silva and his defensive partner in Fortaleza, the latter a superb 30-yard free kick.
Man-of-the-moment James Rodriguez pulled one back from the spot for Colombia in the 80th minute, yet Brazil held on to book a semi-final meeting with Germany.
However, Brazil will head into the last four with a series of worries, most notably the suspension of Silva and an injury to star man Neymar.
Silva picked up a needless yellow card in the second period, while Neymar was carried off on a stretcher late on to put a dark cloud over Brazil's win.
The duo's potential absences, combined with another workmanlike showing, will do little to dispel claims from the likes of Diego Maradona that Brazil are not good enough to win the tournament.
Much had been made of Brazil's perceived mental frailties ahead of the match, with players coming under fire for crying during renditions of the national anthem and Scolari's decision to call in a sports psychologist grabbing headlines.
Silva was the hero, silencing his critics by sneaking behind Carlos Sanchez and kneeing home at the back post from close range after Colombia fatally failed to clear Neymar's corner.
Behind for the first time in the competition, Colombia found an immediate response as Juan Cuadrado's effort was deflected just wide.
Yet it was still Brazil looking the more purposeful, Hulk twice forcing good stops from David Ospina.
Colombia wasted a James-led four-on-two counterattack in between those saves, with the game's end-to-end nature making for a thrilling spectacle.
The second period did not assume the same pace, Jose Pekerman's side failing to find the fluency that has been their calling card.
Indeed, the biggest moment of those exchanges came when Silva collided with Ospina as the Colombia shot-stopper attempted to kick from his hands, the subsequent yellow card ruling him out of the next phase.
With chances few and far between, it took a moment of brilliance from Luiz to leave Brazil on the brink of the last four after 69 minutes.
James, so often Colombia's star, crucially brought down Hulk 30 yards from goal, paving the way for Luiz to send a stunning strike beyond Ospina.
But just when it seemed a nation could calm down, there was a twist in the tail when Julio Cesar felled Carlos Bacca in the area.
The goalkeeper was fortunate not to be shown a red, although that had little impact on tournament top-scorer James - who continued his record of scoring in every game when he coolly sent Cesar the wrong way.
As if the tension was not palpable enough, many in attendance were left with heads in hands late on as Neymar was taken from the field after being caught by Zuniga in the dying minutes.
Brazil defended defiantly during injury time to seal their semi-final spot against Germany and continue their quest for a sixth world title.