She replaces Taylor Swift as the youngest person ever to win the award.
"I joke around a lot at these things, but I genuinely want to say I'm so grateful," said the singer.
Eilish triumphed in all of the Grammys "big four" marquee categories - song of the year, record of the year, album of the year and best new artist.
She is the first person to achieve the feat since Christopher Cross in 1981.
Grammys 2020: Big speeches and emotional moments
In Pictures: Highlight from the 2020 Grammys
Grammy Awards 2020: List of nominees and winners
Her elder brother, Finneas O'Connell, also picked up producer of the year for his work on Eilish's album.
He said the record had been made at home because "I'm the most creative where I'm most comfortable," adding: "It's a huge honour to be given a Grammy for making home-made cookies."
Eilish appeared to be overwhelmed by the extent of her domination of the awards. Accepting the album of the year prize, she turned to fellow nominee Ariana Grande and said: "Can I just say that I think Ariana deserves this?" (Grande waved off the comments, ceding the prize back to the winner).
Earlier, on the red carpet, the singer said she felt like an impostor.
"I feel like I'm not supposed to be here," she joked. "I feel like they accidentally let in a fan."
But the teenager has re-written the rules of pop over the last 12 months, creating ominous, unsettling songs that disrupt typical song structures and lure listeners down dark sonic avenues.
"We didn't make this album to win a Grammy," said Finneas, joining his sister on stage.
"We wrote an album about depression and suicidal thoughts and climate change and being a bad guy, whatever that means.
"And we stand up here confused and grateful."
Kobe Bryant tributes
Other big winners on the night included Lil Nas X, who won video of the year for Old Town Road, and Lizzo, who scooped three prizes including best pop solo performance for her breakout hit, Truth Hurts.
The singer also opened the ceremony and, along with host Alicia Keys, paid tribute to basketball star Kobe Bryant, who died earlier in the day.
"Tonight is for Kobe," Lizzo announced as the show began, before singing the lines, "I'm crying 'cos I love you".
After her performance, Keys, who was hosting the show, walked solemnly to the stage and asked the audience to remember Bryant's family.
"I would like everybody to take a moment and hold them inside of you and share our strength and our support," she said.
Several other performers, including Lil Nas X, DJ Khaled and Run-DMC, paid tribute to the star during the ceremony by holding his jersey aloft.
Stand-out moments came from hip-hop auteur Tyler, The Creator, who performed in the middle of a burning house; and pop singer Demi Lovato, who sang live for the first time since a suspected overdose two years ago.
Ariana Grande, who refused to attend last year's Grammys after a public clash with the show's producer, Ken Ehrlich, made amends with a lavish, risqué and heavily bleeped-out medley of her songs Imagine, Seven Rings, and Thank You, Next, backed by an orchestra.
British artists were thin on the ground, but the Chemical Brothers won two awards, including best dance album for No Geography.
Dua Lipa, who won best new artist last year, was also on hand to pass the baton to Billie Eilish.
She used her speech to champion gender diversity in the music industry, which is still overwhelmingly dominated by men.
"There are so many stellar female producers, artists, songwriters [and] engineers," she said.
"If you're in the business and you're hiring, raise your sights to the amazing, talented women out there because we deserve a seat at every table."