A brilliant goalkeeping display from 40-year-old Gabor Kiraly kept Hungary in with a chance – the pick of his saves seeing him push Kevin De Bruyne's free-kick on to the crossbar.
Hungary pushed for a leveller in the second half, but Hazard's late intervention ended any hope of a comeback.
Hazard set up substitute Michy Batshuayi to score with his first touch, before netting with an excellent solo strike himself as two goals in as many minutes ensured Belgium's progression.
Another substitute, Yannick Ferreira Carrasco, then slotted past Kiraly in stoppage time to add gloss to the scoreline.
After losing Laszlo Kleinheisler to injury in the warm-up, Hungary made a torrid start and fell behind inside 10 minutes.
Kiraly had already denied Romelu Lukaku and De Bruyne early on, but could do nothing when the Manchester City playmaker's exquisite free-kick was converted by Alderweireld's thumping header from eight yards.
De Bruyne should have made it two but, having been released by Lukaku, he could not beat the goalkeeper as Kiraly saved with his legs.
Balazs Dzsudzsak forced Thibaut Courtois, who had earlier got away with an embarrassing slip, into a decent save at the other end, but Kiraly was soon called upon again to keep Hungary in the game.
The veteran rose superbly to tip a brilliant De Bruyne free-kick on to the bar and stretched himself to make a pivotal stop from Dries Mertens later in a first half that saw Belgium rack up 16 shots on goal – a Euro 2016 record.
Hungary, though, had Courtois worried when Gergo Lovrencsics drove a powerful strike inches over and Dzsudzsak sent an effort narrowly wide.
Kiraly was at it again to keep out Hazard early in the second half, but Hungary were thwarted by a one-handed Courtois save when Adam Pinter's shot deflected menacingly off Alderweireld.
Hungary continued to push for an equaliser and Roland Juhasz drilled an attempt inches wide from a tight angle, but Bernd Storck's side were not creating many clear chances despite their possession.
Thomas Vermaelen was booked to rule him out of the Wales game, before Hazard produced two fine moments to end the match as a contest.
After 78 minutes, the Chelsea winger made great ground on the left and laid his cross on a plate for Batshuayi to slot home 95 seconds after coming on.
A few moments later, Belgium's swift counterattack led to De Bruyne feeding Hazard, who eluded three defenders as he cut in from the left and swept a trademark finish into the far corner from 18 yards.
A three-goal margin was already harsh on Hungary, but Carrasco made it worse when he collected Radja Nainggolan's pass and converted with his left foot as Belgium continued their best run at a European Championship since 1980.
Key Opta Stats:
- Belgium became the first team to score four goals in a match at Euro 2016.
- Kevin De Bruyne has scored or assisted 12 goals in his last 12 games for Belgium (5 goals, 7 assists).
- Gabor Kiraly made 10 saves versus Belgium; more than any other goalkeeper in a match so far at Euro 2016.
- There were just 94 seconds between Belgium’s second and third goals of the game.
- This was Belgium’s biggest ever win at a major tournament.
- Michy Batshuayi scored with his first touch for Belgium, after coming on as a second half substitute