It was a miserable night for Hungary on the field and the country's Football Federation is likely to face a FIFA investigation after crowd misbehaviour and reports of racial abuse of two of England's Black players.
England showed no signs of a hangover from their Euro 2020 final defeat by Italy with an excellent second-half display from Gareth Southgate's side.
"It is a real tough place to come, we knew it was going to be a difficult game and such a big game in the group as well, we knew if we won here it puts us into a great position and almost one foot into the World Cup," captain Harry Kane said.
"It couldn't have been better than 4-0 away from home, a great night for us," he added.
Sterling was the provider for England's second eight minutes later, when his deflected cross was headed in by a stooping Kane.
Declan Rice added the fourth, three minutes from the end, with a shot from outside the box which got through the body of Gulacsi who should have kept the effort out.
Gulacsi had made a series of saves from Kane with the England striker lacking his usual clinical touch.
Hungary, already facing two games behind closed doors in UEFA competition after incidents at their national stadium during Euro 2020, could face sanctions from FIFA after fans threw objects, mostly beer cups, at England players on two occasions. A flare was also thrown on to the pitch.
British broadcaster ITV said racist "monkey noises" were aimed at Sterling and England substitute Jude Bellingham from the hardcore "ultras" behind the goal.