A Fernandinho own goal and a brilliant Kevin De Bruyne strike gave Belgium a spot in the last four for the second time after 1986 and sees Brazil eliminated in the quarterfinals for the third time in the last four World Cups.
Brazil dominated possession and carved out chance after chance but, despite the best efforts of Neymar, were unable to find the net until substitute Renato Augusto headed home Philippe Coutinho's cross 14 minutes from time.
The game began with Belgium on the front foot - the likes of Nacer Chadli, Lukaku, Eden Hazard and De Bryune combined to put pressure on Brazil's backline, thus leaving the impression that Brazil will be unable to keep up with Martinez' side during the entire tie.
Perhaps it was the belief instilled in his players by Spanish manager Roberto Martinez and his assistant coach Thierry Henry, a former World Cup victor with France. The other two quarterfinal matches will be played Sunday.
Belgium then scored from their first corner of the game in the 13th minute.
Brazil's centre-half partnership of Miranda and Thiago Silva have looked steady so far, although they are yet to face a team with the Belgium's level of attacking talent.
"Maybe it's this sort of match that we needed for the future", said Hazard, the captain.
Belgium refused to fold, however, in the enthralling closing stages, ensuring Roberto Martinez's side reached the World Cup semi-finals for only the second time in their history. Individual talent is a big weapon of both teams, we rely on one-on-ones, being able to score, open-play football.
For Belgium, defender Jan Vertonghen and De Bruyne are carrying yellow cards as well.
Neymar, who was criticized ahead of the game for having spent almost 14 minutes on the floor cumulatively in Brazil's first four games, flung himself down but was told to get back up. According to research conducted by Betway, just three of the last 15 shoot-outs have been won by the team that went second.
Roberto Martinez's men will now look ahead to their match against 1998 winners France, having beat five-time holders Brazil at the Kazan Arena to inch one step closer to the most important piece of silverware in all of sport. Brazil was offensively stronger, but in the end we all did it.
It takes some luck to advance from any doe-or-die game, especially at a World Cup.
Minutes later Augusto hung in the air to meet Coutinho's perfectly-weighted cross and direct it past Courtois to spark celebrations among the Brazilian fans in Kazan.