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.
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.”
Smith’s ballad Stay With Me won both song of the year and record of the year – the former recognises the songwriters, while the latter goes to the performer and producer.
Accepting the record of the year trophy, 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 the song on stage with Mary J Blige; while his other awards were best new artist and best pop vocal album.
He lost out on the best album award to long-serving indie troubadour Beck, who was honoured for his twelfth studio LP, Morning Phase. It also won best rock album.