The Red Devils were able to see things through for just a second-ever appearance in the semis, despite being made to hold on in the final 15 minutes following a consolation header from Renato Augusto.
While five-time winners Brazil will head back home, prolonging their wait for glory on the world stage to at least 20 years, Belgium now have a semi-final against France - victorious against Uruguay earlier today - to look forward to in St Petersburg next Tuesday.
Both managers made a couple of changes to their starting lineups, with Marouane Fellaini and Nacer Chadli - key players in the 3-2 comeback win over Japan in the last round - both coming in for Belgium.
Fernandinho was one of the two Brazil players brought into the side, meanwhile, along with fit-again Marcelo, and it was the Manchester City midfielder who gifted the opposition a 13th-minute lead.
The game had been pretty evenly matched up until that point, although the Selecao had done more to warrant a lead as Thiago Silva hit the post from close range when picked out from a corner.
Belgium, eliminated by South American opposition in their last two tournament participations, were holding their own, however, and they doubled their advantage with half an hour played.
If the first goal had a slice of fortune about it, the second was pure skill as De Bruyne arrowed a 25-yarder out of Alisson's reach following some good build-up play from Romelu Lukaku.
A smart near-post flick from Vincent Kompany had to be dealt with at the end of the first half, and De Bruyne came close to a second from a well saved free kick, with the Red Devils looking good value for their lead.
Courtois was also being kept busy in a gripping game of football, though, getting across to push aside another Coutinho curler just moments after saving Marcelo's deflected cross.
Not since the 1938 finals have Brazil recovered from a two-goal deficit at the break to win a World Cup match, which they were desperate to rectify in Kazan.
On came Roberto Firmino for Willian midway through, and the centre-forward so nearly carried on from where he left off against Mexico by getting the smallest of touches to Marcelo's cross, but not enough to divert it over the line.
VAR was consulted but, despite Kompany clearly making some contact with his Manchester City clubmate, the Serbian referee was not asked to review the decision.
Hazard then shot wide at the end of a counter at one end, while substitute Douglas Costa did slightly better by forcing a save from Courtois down low at the other.
Brazil's last win against European opposition in a World Cup knockout tie was against Germany in the 2002 final, and after Douglas Costa had another couple of tame shots saved by Courtois it looked as though that run was about to be extended by a further four years.
Augusto's introduction gave the Selecao renewed vigour, though, taking the Beijing Guoan midfielder a little over two minutes to leave his mark on the game with a fine header.
Coutinho's lofted pass over the top was latched on to by Augusto, who got between two defenders and sent the ball out of Courtois's reach into the bottom corner.
Augusto and Coutinho both wasted good opportunities as the momentum swung completely in Brazil's favour, and the competition's most-successful side so nearly grabbed the late equaliser they craved.
Coutinho's curled shot was heading for the back of the net, only for Courtois to get a touch to the ball at full stretch deep into added time, prolonging Belgium's dream run in Russia.
BRAZIL (4-2-3-1): Allison; Fagner, Thiago Silva, Miranda, Marcelo; Fernandinho, Paulinho (Augusto 73'), Coutinho; Willian (Firmino 46'), Neymar, Gabriel Jesus (Douglas Costa 58')
BELGIUM (3-4-3): Courtois; Alderweireld, Kompany, Vertonghen; Meunier, Witsel, Fellaini, Chadli (Vermaelen 83'); De Bruyne, Hazard, Lukaku (Tielemans 87')