Grande, 23, beat off competition from Justin Bieber, Rihanna and Selena Gomez.
"I can't believe this. I am shaking with nerves," she said. "I hope that I can bring joy to you and make you feel unified and happy."
Former One Direction member Zayn Malik, who quit British boy band One Direction last year to launch a solo career, bested Canadian Shawn Mendes to take the best new artist trophy.
"It's been a crazy year... I didn't expect anyone to still vote for me," said Zayn, who now uses only his first name.
British singer Sting, who last week released his first pop/rock album in a decade, "57th and 9th," sang a medley of his hits and was presented with an honorary award recognising his contribution to the music industry.
Sting said he owed a debt to American rock and roll, adding "While I may be an Englishman in New York, my musical soul is all American."
Bieber performed "Never Give Up" via satellite from a concert in Italy and won awards for best pop/rock male artist, album, song and video.
Lady Gaga, Kendrick Lamar, Nicki Minaj, The Weeknd, Niall Horan, James Bay and John Legend were among the dozens of other performers on the Los Angeles stage.
The American Music Awards were created in 1973 as an alternative to the Grammys.
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