Belgium go into the European Championship quarter-finals for the first time since losing the 1980 final. They were dominant throughout at Stadium de Toulouse after Toby Alderweireld headed them into a 10th minute lead.
The Red Devils wasted a series of great chances to finish the game before Michy Batshuayi, with his first touch as a substitute, Hazard and Yannick Carrasco scored in the last 12 minutes.
Belgium now battle Wales in Lille, near the French-Belgian border, for a place in the semi-finals.
“It’s great, we’re into the quarter-finals after a deserved win,” said Hazard.
“We could have scored more goals, and although their ‘keeper pulled off some great stops we’re not complaining.
“We’re in the last eight, we’re going to Lille and I hope there will be a lot of Belgians there to support us.
“We’re still dreaming of going further in this competition. Now, we have to rest and prepare ourselves for Friday.”
Hungary impressively won their group but if they struck any fear into Belgian hearts it did not show.
From Kevin De Bruyne’s speculative fourth-minute strike that sailed over Gabor Kiraly’s crossbar, the Red Devils were too hot for a Hungary making their return to a major tournament three decades after the 1986 World Cup.
Almost from kick-off, Kiraly was kept busy as Romelu Lukaku, then De Bruyne, after Hazard’s superb delivery, tested the goalkeeper.
When Adam Lang hauled De Bruyne to the ground on the left flank, the Manchester City forward sent in a superb inswinging free-kick that Alderweireld headed past Kiraly at his back post.
With Belgian nerves settled, Hazard in particular dazzled fans with some mazy runs, dribbles, and sublime passes that carved open the Hungarian defence.
But a lack of clinical finishing left Belgium at risk during an encouraging Hungary fightback in the second half.
Kiraly did well to block De Bruyne’s effort with his feet after he powered on to Lukaku’s sweeping pass behind the Hungarian defence.
But De Bruyne headed Lukaku’s perfect chip into Kiraly’s arms. And when Zoltan Gera gave possession away cheaply in midfield, Lukaku’s one-two with Radja Nainggolan saw the Everton striker skew wide as De Bruyne ran in unmarked at the back post.
Dries Mertens won a free kick when he was charged from behind by Tamas Kadar but De Bruyne’s superbly-struck free kick bounced off Kiraly’s crossbar.
The best of Hungary’s few chances came just before the interval when midfielder Gergo Lovrencsics let fly a rasping drive that sailed just over.
Moments later, Thibaut Courtois was at full stretch when Balazs Dzsudzsak fired between Nainggolan’s legs.
Belgium spurned another chance on the stroke of half-time when Mertens’ close-range shot was deflected by Adam Lang.
Immediately after the restart, Kiraly palmed over Hazard’s right-footed drive, but Hungary came fighting back. Striker Adam Szalai fired into the side-netting before rising above the Belgian defence to send a deflected header wide.
Adam Pinter’s deflected lob forced Courtois to tip over, and then Roland Juhasz heading well wide from the resulting corner.
But Belgium responded in style.
Michy Batshuayi came off the bench to punish Hungary when he ran in unmarked after the hour to first-time his shot past Kiraly.
The ‘keeper was then left clutching at thin air when Hazard ran across the edge of the area unchallenged to beat him at his far post.
Carrasco, another substitute, added their fourth in the first minute of added time when he powered down the middle on the counter to beat Kiraly.