The last international act to walk away with three Brits was the Scissor Sisters in 2005.
The singer also made an emotional tribute to the late designer Alexander McQueen as she played a low-key version of her track Telephone on the piano.
Lily Allen kicked off the show at Earls Court, entering the stage on a glittery rocket and performing her hit The Fear.
The star also collected the award for best British female, her first Brit accolade. Not expecting to win, the singer said: "I only wore this wig because I thought it would be hard for the camera men to find me to catch my disappointed face."
JLS picked up best single for Beat Again, their debut song, which went straight to number one last summer. That category was voted for by the public.
Band member Oritse Williams said: "We never ever thought this would happen, you've made our dreams come true."
Dizzee Rascal was named best British male and told the crowd: "It's about time as well."
Watch the winning moments from the ceremony
Kasabian, who performed their hit Fire, took the prize for best British band.
Although most of their acceptance speech was muted, the band managed to thank fans who had supported them since the group formed in 1998.
Florence and the Machine was nominated for three awards and walked away with the coveted award for best British album for her debut Lungs.
Florence Welch, who last year collected the critics' choice award, thanked her fans for buying the album.
She added: "It's been an amazing night. Everyone's been incredible. I've had the best time."
Two special categories were created to celebrate the 30th Brits ceremony - the best Brits performance and the best album since the ceremony began.
The Spice Girls' 1997 performance of their hits Wannabe and Who Do You Think You Are was named the best moment in Brits history, while Oasis' (What's the Story) Morning Glory was named best album.
Oasis frontman Liam Gallagher accepted the award, with no sign of brother Noel, and threw the statuette straight into the crowd.
Jay-Z and Alicia Keys performed Empire State of Mind
Robbie Williams received the honorary award for outstanding contribution to music, ending the show with a live medley of his greatest hits including Let Me Entertain You, No Regrets and Angels.
Singer Ellie Goulding was the recipient of the critics' choice prize, which tips a new act for success in the coming year.
The ceremony also saw performances from Jay-Z and Alicia Keys, who gave a rendition of their track Empire State of Mind. Cheryl Cole, meanwhile, sang Fight For This Love.
Florence Welch and Dizzee Rascal performed a mash-up of their songs Dirty Cash and You've Got the Love.
Comedian Peter Kay held the show together as host, putting his talents to good use while introducing acts and award presenters to the stage.
He described Lady Gaga as "New York's answer to Su Polllard", former Spice Girl Mel B as "Yorkshire's answer to Beyonce", described Kasabian as "Leicester's answer to Aswad" and called Robbie Williams "Stoke on Trent's answer to Shakin' Stevens".
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