A Fernandinho own goal and a Kevin De Bruyne strike gave Roberto Martinez’s men the two-goal cushion at half time.
Renato Augusto halved the deficit 14 minutes from time and though Brazil had late chances to level, Belgium has progressed to the last four for the first time since 1986.
"When you play Brazil you need a tactical advantage," said Martinez who deployed De Bruyne to the right of Romelu Lukaku in a 3-4-3 formation
"Brazil bring that psychological barrier, they've won five World Cups. We had to be brave tactically.”
Belgium skipper Eden Hazard, normally at the hub of Belgium’s attacking sallies in the centre, clung to the left wing as a seemingly deluxe outlet.
"But it wasn't about the tactics,” added Martinez. “It was about the execution of those tactics. If you execute it like the players did, it makes it difficult for the opposition."
Semi-final against France
Belgium’s victory set up a semi-final in St Petersburg on 11 July against France, who beat Uruguay on Friday afternoon.
"It was a test of character and I think we did really well," said De Bruyne, who was voted man of the match.
"I don't care where I play. Wherever I play, I have to contribute," he added.
"I did what I had to do, making sure the team is calm in difficult situations. Making sure we relax on the ball and not take too many risks.
"Sometimes we try to go too quick, but I try to manage the team in any way I can. And I will be the first to fight till the last minutes and I hope my teammates follow my example."
Brazil went into the match on the back of a 15-match unbeaten streak in which they had conceded only three goals.
But that tally was almost doubled within 31 first-half minutes.
Brazilian barrage beaten off
That Belgium repelled the Brazilian barrage after the pause was down to Vincent Kompany’s superb marshalling of the rearguard and flintiness in a midfield bolstered by Marouane Fellaini and Nacer Chadli, whose goals had helped salvage potential humiliation in the last 16 against Japan.
"We played a match in which we had lots of possession, a lot of opportunities, but in terms of effectiveness, Belgium managed to turn their opportunities into goals,” said Brazil coach Tite.
"They weren't superior to us on the pitch but more efficient. That's certain."
Having entered the quarter-final against Brazil as notional underdogs, Belgium has now eliminated the second-best-rated team in the Fifa listings and will face an unpredictable France side.
"It will be a different match,” admitted De Bruyne. “Now we have the time so see how we're going to approach it. "It's the semi-finals and it will be extraordinary. But the whole world will be watching us.
"This only happens once or twice in your career as a footballer. We'll do what we can to win it."