Jeers could be heard from the away end as the England players took a knee to show their support for the fight against racism before kick-off.
Clashes then broke out in the away section, with police using batons in an attempt to control the crowd before appearing to be forced into the concourse.
The Metropolitan Police revealed they had made one arrest on suspicion of a racially aggravated public order offence, and two on suspicion of affray and assault on an emergency worker inside the stadium.
As trouble continued on the concourse and outside Wembley, the police made one further arrest for affray, one for drunk and disorderly and one on suspicion of voyeurism.
Metropolitan Police Events tweeted: "Shortly after the start of tonight's match at Wembley, officers entered the stand to arrest a spectator for a racially aggravated public order offence following comments made towards a steward.
"As the officers made the arrest, minor disorder broke out involving other spectators. Order was quickly restored and there have been no further incidents at this stage."
An FA statement read: "We are aware of an incident in the away section during tonight's FIFA 2022 World Cup qualifier at Wembley Stadium.
"We will be investigating and will report the incident to FIFA."
FIFA also released a statement that said: "We are analysing reports of FIFA World Cup qualifier matches in order to determine the most appropriate action.
"FIFA has a very clear zero-tolerance stance against such abhorrent behaviour in football."
John Stones scored the England goal in a 1-1 draw and said after the game: "I didn't see any of the incidents.
"But if they were showing those sort of signs to what we believe in as a team, as a nation, I haven't got anything good to say about them as supporters, them as human beings.
"We respect everyone and feel we have an incredible platform to voice our word and that's what we do before the game.
"We stand strong together as a team."
Sky Sports' Peter Smith, at Wembley, said: "Someone in the Hungary section held up an anti-taking-a-knee banner as the players made those symbolic gestures.
"In contrast, the England supporters applauded the players' action."
Last month's meeting between the sides in Budapest was marred by racist abuse aimed at England duo Raheem Sterling and Jude Bellingham.
FIFA ordered Hungary to play two home FIFA competition matches behind closed doors, one suspended for two years, following the racist behaviour of their supporters.
When Hungary boss Marco Rossi was asked about the disturbances and it was suggested the fans had, again, let down their national team, he said: "Just like I said before, I don't want to comment on this situation.
"This is not my task and everything that I do say can be interpreted in a different way, so I prefer to make no comment."
Kick It Out is football's equality and inclusion organisation - working throughout the football, educational and community sectors to challenge discrimination, encourage inclusive practices and campaign for positive change.