Brazil's Willian in a past press conference before World Cup game
As Belgium exhausted against waves of Brazilian attacks in the second half, Hazard carried his side up the field, winning free-kicks and relieving the pressure - a sign of his and this side's growing maturity. But it was too late for Brazil to muster an equalizer as Courtois thwarted several efforts to force the game into extra time.
That proved to be Brazil's last chance, and Belgium will march on to their first World Cup semi-final since 1986, where they will face France.
Up next for Belgium in the last four is Didier Deschamps' France, and De Bruyne is ready to grasp a rare opportunity for his country.
"Providing assists is also important, and this year I have provided more than ever". 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.
"Brazil have so much finesse and you know that they can cut you open".
Substitute Renato Augusto gave Brazil hope with just under a quarter of an hour to go with a deft header into the bottom-right corner of the net.
There was to be no repeat of four years ago when Brazil were thrashed 7-1 by Germany in the semi-finals on home soil, however, and they kept their composure despite conceding two goals for the first time in 26 matches under Tite.
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Coutinho badly skewed Brazil's next chance high and wide.
The Brazilians were made to pay for their profligacy in the 13th minute, as Fernandinho diverted a Nacer Chadli corner into his own net.
A devastating counter-attack launched by Manchester United striker Lukaku caused panic, leaving De Bruyne in space and with time to send a powerful drive past Alisson on 31 minutes.
Neymar's bad luck with World Cup quarters continued today with defeat by Belgium; he huffed and puffed, he even rolled over a couple of times but he could not blow down the Belgian defence and he couldn't beat Thibaut Courtois. But Brazil looked impressive against Mexico, and with Neymar getting fitter and more effective with every outing, the Belgians should be concerned.
The Red Devils were everywhere on the pitch in the first half and as the scoreline suggested, they were the better side but tide of the game shifted radically in the second half. "But it is a dream match for our players - they were born to play in a match like this".
Brazil saw two penalty claims in quick succession waved away by Serbian referee Milorad Mazic after Neymar tumbled following contact with Fellaini and Gabriel went to ground after a challenge from Vincent Kompany.
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