Belgium started the half in the best possible way with Alderweireld's opening goal, decent opportunities for both sides followed throughout the rest of the half, with Kevin De Bruyne, Xavier Viera, Balasz Dzsudzsak and Dries Mertens all coming close.
Hazard, Adam Szalai and Adam Pinter had good opportunities in the second period, but goals from Batshuayi, Hazard and Carrasco sealed their passage with ease.
Off to a good start
Marc Wilmot's side were the favourites for this tie, and Romelu Lukaku had a decent opportunity in the ninth minute, but moments after that Belgium managed to take the lead.
It was a simple free-kick routine from the Belgians, as De Bruyne curled a ball into the area and despite the pressure from Lukaku, Alderweireld did well to head home with ease.
De Bruyne should have made it two in the 14th minute, as Lukaku played a great ball into the Manchester City man but despite having time, he could only fire it straight at Gabor Kiraly, who saved with his legs.
As the half went on, Hungary started to establish themselves in the game, but could have been two behind if not for Kiraly. De Bruyne produced an excellent free-kick, and it looked destined for the back of the net, but Kiraly did brilliantly to tip it onto the crossbar.
Hungary then had some excellent opportunities with Viera and Dzsudzsak, with both having Thibaut Courtois stretching, but both were agonisingly wide.
The final opportunity of the half fell to Belgium, and it was one that should of been scored. Hazard fizzed a through ball into the feet of Mertens, but Kilray was on hand once again, to produce a decent save.
Start where they left off
Belgium came flying out of the traps in the second period and almost had their second goal, just seconds after the restart. Skipper Hazard hit a thunderous shot, which look destined to go in but Kilray did brilliantly with the one-handed save.
Hungary continued to grow in the game, and they had two excellent opportunities to equalise.
The first came from striker, Szalai who rose up with Thomas Vermaelen, and the Barcelona man almost turned it into his own net, but luckily for him, it went just wide of the post.
The second came from Pinter, who got the best of Axel Witsel and got his shot off, which took a dangerous deflection from goalscorer, Alderweireld and looked to be creeping under the crossbar, but Courtois managed to tip it over.
Bring on the dragons
Wilmot's side were pretty much under the quash for the majority of the second period, but two goals in quick succession, sealed their place in the last eight.
Batshuayi made an instant impact after his substitution, as he added Belgium's second goal with his first touch. De Bruyne's corner managed to make its way to Hazard, whose burst of pace got him to the by-line to put a ball across the face of goal, and Batshuayi made no mistake as he tapped home.
Hazard went from provider, to scorer just moments later, as a counter-attack ended up at his feet. The Chelsea man had been superb, and was at it again, as he cut in from he wing, across the edge of the area, before smashing it into the bottom corner.
The win was sealed in extra-time as Carrasco scored their fourth and final goal. Another sub showed their impact as Belgium counter-attacked once again through Radja Nainggolan who played a easy ball into the path of Carrasco, who took his time to slot home with ease.