Among her prizes were best female artist, best streaming song for Shake It Off and the ceremony’s top honour, artist of the year.
British star Sam Smith won best new artist – and accepted his trophy using flash cards after having vocal surgery.
But Kanye West was booed as he closed the show with a medley of hits.
Audience members jeered as the rapper’s sisters-in-law, Kendall and Kylie Jenner, introduced his set; and the booing continued during his performance.
Surrounded by pyrotechnics and largely in the shadows, West’s delivery on All Day and Black Skinhead was angry and intense. And while some disapproved, other fans could be seen pumping their fists to the tracks.
TV audiences were also left mystified as the first minute of his set was censored by broadcasters ABC – reminiscent of West’s performance at the Brit Awards earlier this year.
Other artists were more favourably received, with Britney Spears and Iggy Azalea receiving a standing ovation for the first performance of their new song, Pretty Girls.
And Scottish rockers Simple Minds took to the stage to perform Don’t You Forget About Me, marking the 30th anniversary of teen movie The Breakfast Club, which featured the song in its closing moments.
One Direction came away with two trophies, for best group and best touring artist, dedicating the latter to “our brother Zayn” Malik, who left the band in March.
Swift opened the show with the premiere of her star-studded new video, Bad Blood – which features the likes of Ellie Goulding, Selena Gomez, Jessica Alba and Cindy Crawford.
The singer, who has successfully transitioned from country artist to pop star, accepted the prize for best album by thanking the fans who “embraced change”.
“I love you so much, more than I will be able to say to you,” she added.
Smith, who is recovering from throat surgery, sent his acceptance message in a special video.
He held up a series of cards, reading: “Thank you Billboard for this incredible honour. Secondly, thank you to my amazing fans. It’s times like this that I realise how lucky I am to have you in my life.
“I can’t wait to get this surgery out of the way so I can sing my ass off for you again.”
Australian rapper Iggy Azalea won top rap song for Fancy, her collaboration with Britain’s Charli XCX, while John Legend won Top radio song for All Of Me.
Meghan Trainor’s All About That Bass was named top digital song and top Hot 100 song, recognising its US chart performance.
Coldplay won best rock album for Ghost Stories, while Irish singer Hozier picked up best rock song for Take Me To Church.
In tribute to the late singer Ben E King, rock band Imagine Dragons performed Stand By Me to an audience that included the late singer’s daughters and granddaughters.
And presenter Ludacris also gave a touching tribute to blues star BB King, who died on Friday.
Full list of winners
Top artist: Taylor Swift
Top new artist: Sam Smith
Top male artist: Sam Smith
Top female artist: Taylor Swift
Top duo/group: One Direction
Top Billboard 200 artist: Taylor Swift
Top Hot 100 artist: Taylor Swift
Top digital songs artist: Taylor Swift
Top radio songs artist: Sam Smith
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 country artist: Florida Georgia Line
Top rock artist: Hozier
Top Latin artist: Romeo Santos
Top dance/electronic artist: Calvin Harris
Top Christian artist: Hillsong
Top Billboard 200 album: Taylor Swift, 1989
Top soundtrack: Frozen
Top R&B album: Pharrell, GIRL
Top rap album: J Cole, 2014 Forest Hill Drive
Top country album: Jason Aldean, Old Boots, New Dirt
Top rock album: Coldplay, Ghost Stories
Top Latin Album: Enrique Iglesias, Sex and Love
Top dance/electronic album: Lindsey Stirling, Shatter Me
Top Christian album: Lecrae, Anomaly
Top Hot 100 song: Meghan Trainor, All About That Bass
Top digital song: Meghan Trainor, All About That Bass
Top radio song: John Legend, All Of Me
Top streaming song (audio): John Legend, All Of Me
Top streaming song (video): Taylor Swift, Shake It Off
Top R&B song: Pharrell, Happy
Top rap song: Iggy Azalea Feat. Charli XCX, Fancy
Top country song: Jason Aldean, Burnin’ It Down
Top rock song: Hozier, Take Me To Church
Top Latin song: Enrique Iglesias Featuring Descemer, Bailando
Top dance/electronic song: DJ Snake & Lil Jon, Turn Down For What
Top Christian song: Carrie Underwood, Something In The Water
Billboard chart achievement award: Taylor Swift