The chaotic scenes, which occurred shortly before kickoff on Sunday night, were filmed and were shared widely on social media.
It is not known how many people entered the stadium illegally to watch the game.
Metropolitan Police admitted on Monday that they were surprised by the breach, with Assistant Deputy Commissioner Laurence Taylor saying: “I don’t think anyone expected large numbers of people to try to enter the stadium.
“As soon as it became clear that this was happening, the stewards reacted very quickly and the police were deployed to the stadium to prevent the situation from escalating. “
Nearly 50 arrests were made in connection with the final and 19 officers were injured as they faced the “volatile” crowds, the force said.
Maguire, who plays for Manchester United, said the sun that his father Alan and his agent Kenneth Shepherd were trampled by people who overrun security and ran into the ground through a disabled entrance.
Speaking about his father, the 28-year-old said: “It wasn’t a great experience, it shook him. But he was lucky because in every game he played he had my nephew or one of my children on his shoulders.
“So I’m grateful that it didn’t happen because it could have been a really serious moment. “
He added that his father had difficulty breathing because of his injuries. “Things could have been a lot worse, but we have to make sure that doesn’t happen again,” he said.
Following the chaos at Wembley, UEFA initiated disciplinary proceedings against the English Football Association. Four of the charges involved the game itself, including the throwing of objects on the field and the invader who interrupted the game.
In addition to talking about the stampede, Maguire said the sun that he wants social media companies to do more to crack down on hate speech, after three of his teammates – Marcus Rashford, Jadon Sancho and Bukayo Saka – suffered racial violence online after they missed their shots on goal in the penalty shoot-out on Sunday.