The record also earned the band the best rock album title, while they won song of the year and best rock performance for Sometimes You Can't Make It On Your Own.
Their track City Of Blinding Lights triumphed in the best rock song category.
U2's success upstaged Mariah Carey, who with eight nominations had been billed of the comeback star of the night, but only picked up three awards in the end.
"I don't know what to say; this is really a big, big night for our band," frontman Bono told the 48th annual awards in Los Angeles.
The Chemical Brothers led the British winners, picking up the best dance recording award for Galvanize and best electronic/dance album for Push The Button.
Cartoon band Gorillaz, headed by Damon Albarn, won best vocal pop collaboration for their single Feel Good Inc, featuring De La Soul.
But Sir Paul McCartney, who had been nominated in three categories, went home empty-handed despite a rousing rendition of the Beatles' Helter Skelter, his first at the Grammys.
Gorillaz, who kicked off the 48th annual awards in Los Angeles with a "performance" with Madonna, lost out in three categories.
The leotard-clad 47-year-old joined the characters on stage for Feel Good Inc, which was mixed into her hit Hung Up.
U2's Sometimes You Can't Make It On Your Own is a tribute Bono wrote for his late father.
"I want to thank my father Bob for giving me voice and a bit of attitude," he said as he accepted the song of the year trophy.
He added: "If you think this is going to go to our head, it's too late."
Accepting the best rock album award, guitarist The Edge said it was "amazing".
"It means a hell of a lot to us," he said. "An even more precious gift than the awards is the gift that you've all afforded us to continue to make our music."
The Grammys, handed out by the National Academy of Recording Arts and Sciences, are regarded as the most prestigious music prize in the US.
U2's clutch of awards left Coldplay disappointed. They had been nominated in the rock album, rock song and rock performance categories but were pipped to the post by the Irishmen every time.
And the Chemical Brothers' success came at the expense of fellow Brit Fatboy Slim.
Keane, Rod Stewart, Eric Clapton, Seal and Franz Ferdidand also failed to win awards.
David Bowie and Cream were honoured with lifetime achievement awards and there was a best musical show album gong for Monty Python's Spamalot.
An increasingly ecstatic Bono told the audience: "There's the possibility with rock music... that it might just amount to something a little more than entertainment on an occasion and that you might be able to communicate some honest feelings."
Picking up the best album title, he paid tribute to those they had beaten to the award - Carey, Sir Paul, Kanye West and Gwen Stefani.
"Mariah, you sing like an angel, you're really something else," he said.
Earlier he explained that the title How To Dismantle An Atomic Bomb had been inspired by his relationship with his father.
"When he died it set off a kind of chain reaction in me," he said.
"I've been shouting about him and giving off about him and complaining about him and screaming about him for the last few years and maybe tonight is the time to stop."
"It's really a great, great moment for me personally."
U2 performed their hit Vertigo, and collaborated with R&B queen Mary J Blige for their classic, One.
Carey ended her 16-year Grammy drought, but her three awards - including best contemporary R&B album for The Emancipation of Mimi - were all in the pre-telecast ceremony, meaning viewers saw her perform but never heard from her.
One of the best-selling artists of all time, she had not won a Grammy since her first two as a fresh-faced ingenue in 1990.
She lost twice to U2, once to Green Day for record of the year and once to former American Idol Kelly Clarkson for best female pop vocal performance.
The surprise of the show was the appearance of Sly Stone, the mercurial, psychedelic pioneer who disappeared from the music scene decades ago and had not performed in public since 1993.
Near the end of an all-star tribute, Stone, 61, emerged sporting a pale Mohawk and made his way through one of his biggest smashes, I Want To Take You Higher.
Although the tribute was planned, many had not expected him to turn up.Reuse content