They were named artists of the year and also took home album of the year and favourite pop/rock album.
"This has been a real perfect day... America feels like a second home to us," said Liam Payne.
Katy Perry, who did not appear at the ceremony in Los Angeles, also won three awards, including one for best single for Dark Horse featuring Juicy J.
The awards, many of which are decided by a fan vote, also saw Taylor Swift speak out again in defence of traditional models of music distribution.
The singer, who was presented the Dick Clark award for excellence by Motown musician Diana Ross, thanked fans for acknowledging her campaign.
"What you did by going out and investing in music and albums is you're saying that you believe in the same thing that I believe in - that music is valuable and that music should be consumed in albums, and albums should be consumed as art and appreciated," she said.
After the release of 1989, Swift removed all of her albums from Spotify, saying the streaming service devalued music and did not pay artists enough.
Iggy Azalea beat the likes of Eminem and Drake to the title of favourite rap/hip hop artist.
"This award is the first award I've ever won in my entire life, and it means so much to me that it is for best hip hop because that's what inspired me to move to America and pursue my dreams, and it's what helped me when I was a teenager to escape and to get through my life and to better times," the 24-year-old said.
"And it means so much that I can stand here against Eminem and Drake and come out with this."
Calvin Harris and Sam Smith took home favourite artist awards in the EDM and pop/rock categories. Frozen was named best soundtrack.
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One Direction have topped the winners list at the American Music Awards, taking home a total of three prizes.