Roberto Martinez’s side turned on the style in Kazan to clock up their fifth straight win of the tournament and ensure the semi-finals will be contested by four European sides for the first time since 2006.
Brazil had conceded just one goal in their past four games in Russia but were behind early when Nacer Chadli’s corner struck Fernandinho’s arm and flew into the net, the 10th own goal of this World Cup.
Fernandinho’s Manchester City team-mate Kevin de Bruyne doubled Belgium’s lead with a sublime arrowed finish from 20 yards after Romelu Lukaku’s powerful run.
In an entertaining and open game, Brazil hit the post when the match was goalless – Thiago Silva missing in front of a gaping net after Neymar’s corner – before substitute Renato Augusto gave Brazil hope with a headed goal after Philippe Coutinho’s delicious pass.
However, the South Americans became the latest heavyweight to depart the competition after holders Germany and other past winners Argentina, Spain and Uruguay.
Impressive Belgium will now face France, 2-0 winners over Uruguay earlier on Friday, in a semi-final in St Petersburg next Tuesday, 10 July (19:00 BST).
The other semi-final in Moscow on Wednesday 11 July will be contested between the winners of Saturday’s quarter-finals featuring Sweden and England (15:00 BST) and Russia and Croatia (19:00 BST).