The 23-year-old singer also won country album of the year for "Red" and favorite female artist in both the country and the pop/rock categories.
"Winning artist of the year for the third time is kind of a little mind-numbingly unbelievable. It's a little bit hard to absorb it all at once. It's amazing, and the fact that it's fan voted makes it that much better," Swift told reporters.
Swift didn't perform, but she was a constant presence on stage and the camera alighted on her as she danced to stars like Kendrick Lamar and Luke Bryan.
Justin Timberlake won three AMAs, including favorite male artist in both the pop/rock and soul/R&B categories.
Top-selling British boy band One Direction won two awards: favorite pop/rock band, duo or group, and favorite pop/rock album for "Take Me Home."
The band caused quite a commotion on stage, but their very vocal fans almost caused a scene before the show began. Police had to shore up a fence outside the Nokia Theatre in Los Angeles when the eager 1D fans heard their favorite band had arrived.
Another notable red carpet arrival: Lady Gaga and a puppeteered horse. Gaga also performed on the show with R. Kelly. It was a night that was really more about performances than awards.
Katy Perry opened the show looking like a princess out of a classic Japanese painting. Dressed in a traditional Japanese dress, Perry's show-starting performance of "Unconditionally" included dozens of colorfully clad dancers waved fans, shadow danced in front of rice-paper screens and played the drums.
Sixteen different artists took the stage. Miley Cyrus sang "Wrecking Ball" with an animated cat behind her, and country star Luke Bryan got the people in the audience on their feet.
Nickelodeon star Ariana Grande was named favorite new artist, Florida Georgia Line won single of the year for "Cruise" with Nelly in something of a surprise, and Luke Bryan won favorite male country artist.
Rihanna and her mother took center stage midway through the show when Monica Fenty presented her pop star daughter with the first Icon Award.
Jennifer Lopez nailed the elaborate dance numbers during her tribute to Salsa queen Celia Cruz, taking two dozen dancers through their paces while wearing a shimmering silver dress.
Big winner Swift said she views the AMAs as an opportunity to see a great concert.
"Oh my God! The Great Big World and Christina Aguilera performance, I was almost crying; it was so emotional," said Swift. "I just thought that there were so many good ones; Imagine Dragons killed me."
The Associated Press contributed to this report.