Smith was rewarded for being the only solo artist to sell more than a million albums in both the UK and US in 2014.
Pharrell Williams, Beyonce and Rosanne Cash, daughter of country legend Johnny, won three awards each.
Beck won two, including best album. Other British winners included Clean Bandit with Jess Glynne and Aphex Twin.
Madonna, Katy Perry, Rihanna, Kanye West and Sir Paul McCartney were among those performing at the Los Angeles ceremony.
But the night belonged to Sam Smith, who has cemented his place as one of pop’s biggest stars just nine months after the release of his debut album In the Lonely Hour
Accepting the trophy for record of the year, he said: “I want to thank the man who this record is about, who I fell in love with last year.
“Thank you so much for breaking my heart because you got me four Grammys.”
Smith also performed his hit Stay With Me on stage with Mary J Blige.
Earlier, he said: “Before I made this record, I was doing everything to try to get my music heard… I tried to lose weight and I was making awful music. It was when I started to be myself that the music flowed.”
As well as record and song of the year for Stay With Me and best new artist, Smith won best pop vocal album.
He lost out on the best album award to long-serving indie troubadour Beck, who was honoured for his 12th studio LP Morning Phase. It also won best rock album.
“We made this record at my house for the most part, so I would like to thank my kids for letting me keep them awake a little bit longer,” he said.
- Album of the year: Morning Phase – Beck (above)
- Record of the year: Stay With Me – Sam Smith
- Song of the year: Stay With Me – Sam Smith
- Best new artist: Sam Smith
- Best pop vocal album: In the Lonely Hour – Sam Smith
- Best pop solo performance: Happy (live) – Pharrell Williams
Pharrell’s Girl won best urban contemporary album, while Happy won best music video and best pop solo performance. He performed that hit on the night with pianist Lang Lang and composer Hans Zimmer.
Drunk In Love, Beyonce’s collaboration with her husband Jay-Z, was named best R&B performance and best R&B song. Beyonce also won best surround sound album.
The other triple winner, Rosanne Cash, picked up the best American roots performance and song awards for A Feather’s Not A Bird and best Americana album.
Among the British artists, Clean Bandit and Jess Glynne won best dance recording, while Aphex Twin won best dance/electronic album for Syro, his first album for 13 years.
Ed Sheeran, who had three nominations, went home empty-handed, but did join ELO’s Jeff Lynne on stage to perform at the ceremony.
Other performers on the night included Sir Paul McCartney, Kanye West and Rihanna, who teamed up to play their joint composition FourFiveSeconds.
Madonna was joined by half-naked male dancers wearing glittering masks and horns to perform Living For Love, from her new album Rebel Heart.
There were also duets between Jessie J and Sir Tom Jones; Hozier and Annie Lennox; and Lady Gaga and Tony Bennett.
The ceremony departed from pure entertainment for a segment in which US President Barack Obama sent a recorded video message in support of a campaign against domestic violence.
An abuse victim from Texas then delivered a poignant message about her own experiences.