Swift also won top Billboard 200 album and top female artist Sunday night.
"Oh, this is such a good night!" she said after winning the top prize, besting One Direction, Ariana Grande, Sam Smith and Ed Sheeran. "Early this morning I watched my brother Austin graduate from Notre Dame ... (and) to anyone who's graduating this year ... this is for you."
"This is kind of a graduation for us," Swift said to the audience of top musicians.
West, however, was booed during his smoky performance of "All Day."
"And now a doctor," Kylie Jenner said while introducing the rapper, who earned an honorary doctorate from the Art Institute of Chicago.
Swift kissed her rumored boyfriend Calvin Harris on the cheek and gave him a hug at the MGM Grand Garden Arena in Las Vegas. Swift, who sat next to the DJ-producer-singer, kicked off the awards by debuting her action-filled music video for "Bad Blood," which stars Selena Gomez and Lena Dunham and showcases Swift as a fighting vixen.
The video opened with Swift and Gomez fighting men in an office and then fighting each other. It ended with Swift's and Gomez's armies going head to head. Other stars in the clip included Mariska Hargitay, Ellen Pompeo, Jessica Alba and Ellie Goulding, and Hayley Williams of Paramore.
Swift swapped her first two verses with new ones from rapper Kendrick Lamar for "Bad Blood," a song rumored to be about Swift's friendship with Katy Perry.
"I am so excited I got to show the world," Swift told the audience.
One Direction, which recently downsized to four members, won the night's first award for top duo/group.
"There is one more person to share this with, and that's our brother Zayn," Liam Payne said of Zayn Malik, who left the group.
The boy band also won top touring artist.
Iggy Azalea won top rap song for "Fancy" as her NBA player boyfriend, Nick Young, watched her with emotion. Azalea also won top streaming artist and rap artist.
Van Halen was the night's first performance and rocked the crowd with its 1984 hit "Panama." Mariah Carey sang her first hit, "Vision of Love," which was pitchy. She followed it with new single, "Infinity," where she also struggled. She ended with a strong high note, though.
Britney Spears was energetic when she performed her new single, "Pretty Girls," in a sheer black cat suit alongside Azalea. Swift and Zendaya sang along as Nicki Minaj mainly danced, barely singing or rapping live and relying on a backing track during "Hey Mama" and "The Night Is Still Young."
Meghan Trainor performed a duet with John Legend and won hot 100 song for "All About That Bass." She was teary and shaky onstage, as was her friend and collaborator Charlie Puth, who performed "See You Again" with Wiz Khalifa earlier in the show.
Other performers included Pitbull with Chris Brown and Little Big Town with Faith Hill.
Hosts Ludacris and Chrissy Teigen offered some jokes Sunday night: The rapper described why he's a good host, namechecking his music and acting credits. Teigen, a model, replied, "And I bang a musician" as her husband, John Legend, stood up.
Legend later won top radio song for his No. 1 hit, "All of Me."
"I have to thank, of course, my beautiful co-host, my lovely wife, Chrissy Teigen," he said. "I am so proud that we won this award together tonight."
Ludacris offered some words about the late B.B. King and his renowned career while the camera panned to a guitar on an empty stage.
Swift was the lead nominee with 14 nominations. Smith, who recently had vocal surgery, didn't attend the show. He won top male artist and wrote his thank you speech on cue cards, which aired via video.
Here's a list of the winners:
Top Billboard 200 Album: Taylor Swift, "1989"
Top R&B Album: Pharrell Williams, "G I R L"
Top Rap Album: J. Cole, "2014 Forest Hills Drive"
Top Country Album: Jason Aldean, "Old Boots, New Dirt"
Top Rock Album: Coldplay, "Ghost Stories"
Top Christian Album: Lecrae, "Anomaly"
Top Latin Album: Enrique Iglesias, "Sex and Love"
Top Soundtrack: Frozen
Top Artist: Taylor Swift
Top Female Artist: Taylor Swift
Top Male Artist: Sam Smith
Top Duo/Group: One Direction
Top New Artist: Sam Smith
Top Billboard 200 Artist: Taylor Swift
Top Hot 100 Artist: Taylor Swift
Top Country Artist: Florida Georgia Line
Top Digital Songs Artist: Taylor Swift
Top Touring Artist: One Direction
Top Social Artist: Justin Bieber
Top Streaming Artist: Iggy Azalea
Top R&B Artist: Pharrell Williams
Top Rap Artist: Iggy Azalea
Top Rock Artist: Hozier
Top Latin Artist: Romeo Santos
Top Dance/Electronic Artist: Calvin Harris
Top Christian Artist: Hillsong United
Billboard Chart Achievement Award (Fan-Voted): Taylor Swift
Top Hot 100 Song: Meghan Trainor, "All About That Base"
Top Radio Song: John Legend, "All of Me"
Top Rap Song: Iggy Azalea ft. Charli XCX, "Fancy"
Top Digital Song: Meghan Trainor, "All About That Bass"
Top Streaming Song (Audio): John Legend, "All of Me"
Top Streaming Song (Video): Taylor Swift, "Shake It Off"
Top Dance/Electronic Song: DJ Snake & Lil Jon, "Turn Down For What"
Top Christian Song: Carrie Underwood, "Something in the Water"
Top Latin Song: Enrique Iglesias feat. Descemer Bueno & Gente de Zona, "Bailando"
Top Rock Song: Hozier, "Take Me to Church"