Hazard, who plays for German side Borussia Dortmund, hit a swerving shot from 25 yards which deceived Rui Patrício in the Portugal goal and knocked the reigning champions out of the tournament.
Portugal had opportunities to draw itself level, with Raphaël Guerreiro striking the post and Ruben Dias forcing a good save from Thibaut Courtois in the second half.
However, it proved all in vain, as Belgium advanced to the quarterfinals where it will face Italy in Munich on Friday.
“In these kinds of games you have to grab your chances with both hands,” Hazard said afterwards.
“And voila, the goalkeeper expected me to choose the other corner, so the ball went in. I am happy I could score for my country!”
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