Luiz doubled the Brazilian lead with his spectacular free kick that crashed into the top right of the net in the 69th minute while skipper Thiago Silva's first half strike on a Neymar corner kick had put the hosts ahead.
Colombia pulled one back when James Rodriguez converted a penalty in the 80th minute but Brazilians held nerves in the tense remaining time to set up a last-four clash with European powerhouse Germany, who ousted old rivals France 1-0 in Rio De Janeiro early in the day.
The hosts though have some worries going into Tuesday's showdown as Brazilian skipper Silva will miss the last-four clash due to double booking and Neymar's fitness is under cloud after he was carried off on a stretcher following a rough challenge from Colombian defender Juan Camillo Zuniga late in the second half.
Neymar, kneed on the back by Zubiga, was taken to a hospital to undergo tests.
In a fast and frenetic quarterfinal clash, Brazil came up with its best show so far in the home tournament even as Colombia put up a spirited fight.
Young Colombia striker Rodriguez was gutted as he left the field in tears before exchanging his shirt with Luiz, who consoled the rival player.
Rodriguez scored his sixth goal of the tournament but his penalty was not enough to keep his side alive in the tournament.
Silva struck as early as in the seventh minute when Neymar's corner was missed by Luiz in the middle but the home skipper smashed in with knee.
Silva became the first captain to score for Brazil in a World Cup game in 20 years and it was Brazil's quickest World Cup goal since Ronaldo scored in the fifth minute against Ghana in 2006.
It was slack defence from Colombia as Silva was allowed to drift in at the back post.
Two minutes later, Hulk struck powerfully but Colombia keeper David Ospina got down just in time to thwart the challenge. The burly Hulk was at it again as he initiated a flurry of attacks but Ospina denied him once again and the hosts took the break with a 1-0 lead.
Surprisingly, the pace of the game dropped a bit in the second-half and there were fewer of chances as the play was restricted to mid-field.
Brazilian coach Felipe Scolari was frustrated when Silva collided with Ospania and was booked.
However, Luiz with his extraordinary free kick brought the cheers back.
A 2-0 lead had settled the home players but they were up to challenge again when Colombian substitute Carlos Bacca was brought down by Julio Cesar inside the penalty area.
Rodriguez converted the resultant penalty, sliding the ball left as Cesar dived on the right. A charged up Colombia tried hard to get the equaliser but it never came.
Earlier, in Rio De Janeiro, Mats Hummels' first half header proved decisive as Germany reached FIFA World Cup semifinals for the fourth successive time with a 1-0 win over fellow European giants France.
The contest between the two European powerhouses lacked spark even as Germany keeper Manuel Neuer was also outstanding under the bar as he made two spectacular saves in each half to deny France the equaliser.
Germany had more possession but it was France, who had first strike in the rival goal post in the seventh minute but Karim Benzema could not provide the finishing touch as he hit a volley wide on a pass from Mathieu Valbuena.
The Germans were clearly the more dominating side and they were rewarded for their effort as Hummels headed home a superb free-kick from the left taken by Toni Kroos.
Hummels jumped and out-powered French defender Raphael Varane, nodding home the lead in the 13th minute.
The Germans continued to enjoy great possession thanks to to some superb work by their midfield trio of Toni Kroos, Sami Khedira and Bastian Schweinsteiger.
A rattled France upped their game and swung to action. Again it was Mathieu Valbuena from the left, making a superb effort but Germany keeper Neuer stuck his left arm out made a great save.
Germany also survived the resultant corner as Mahadou Sakha headed over the cross bar.
Veteran Germany striker Miroslav Klose was handled pretty badly as he was taken down a number of times by the French defenders. Klose also sought a penalty on one such tackle but the referee did not oblige.
France improved near the break. Benzema started to find holes in German defence. The Real Madrid player rushed from the left took a pass from Paul Pogba but his feeble shot landed safely in Manuel Neuer's arms.
The French side was different in the second half, which was far from the German domination. Antoine Griezmann and Valbuena were staying up as close to Benzema as possible.
Germany midfielder Sami Khedira was shown the first yellow card of the match as he brought down Griezmann, nine minutes into the second half.
The French side raided more even as efforts did not bear fruit. Their midfielder Blaise Matuidi rushed forward with a superb sprint but his cut-back from the left wing was well intercepted by a sliding Bastian Schweinsteiger.
German keeper Neuer came to action first time when he easily picked a header from Raphael Varane.
A couple of substitutions followed as Chelsea player Andre Schurrle replaced Klose for Germany while France brought in Arsenal centre-back Laurent Koscielny in place for Mamadou Sakho.
France did succeed in creating a few half chances but never got that equaliser. Schurrle had a chance to double the German lead but his low shot was was blocked by Raphael Varane.