Eilish, an 18-year-old newcomer, won for her debut studio album "When We All Fall Asleep, Where Do We Go?" and her single "Bad Guy" was named record of the year and song of the year.
The Los Angeles singer recorded the album with her brother Finneas in his bedroom.
"We didn't make this album to win a Grammy. We wrote about depression and suicidal thoughts and environmental change," he said as the pair accepted the awards in Los Angeles on Sunday. "We stand up here confused and grateful."
"Wow! So many other songs deserve this," said Eilish. "I genuinely want to say I am so grateful and I am so honoured to be here. I grew up watching all of you."
Fellow fresh face Lizzo won three Grammys, and gay country rapper Lil Nas X won two for his viral "Old Town Road" collaboration with Billy Ray Cyrus.
The Grammy Awards opened with a dedication to basketball star Kobe Bryant, who died in a helicopter accident outside Los Angeles earlier in the day.
"Tonight is for Kobe" said Lizzo, kicking off the show with a medley of the title song from her album "Cuz I Love You" and hit single "Truth Hurts."
The Grammy Awards took taking place at the Staples Center in Los Angeles, the home of Bryant's team the Los Angeles Lakers.
"We are literally standing here heartbroken in the house that Kobe Bryant built," said Grammy host Alicia Keys. "We never imagined in a million years we'd have to start the show like this."
Lizzo won three of her leading eight nominations, including best urban contemporary album for "Cuz I Love You," and pop solo performance for "Truth Hurts," beating established artists like Beyonce, Ariana Grande and an absent Swift.
Gay black rapper Lil Nas X (20) wearing a cowboy hat and silver lurex suit, and country singer Billy Ray Cyrus brought the house down with a kitschy performance of their viral collaboration "Old Town Road," with contributions from stars ranging from K-Pop band BTS to young yodeller Mason Ramsey.
"He told the world he was gay and overnight he became an inspiration and a role model for hundreds of young people around the world," comedian Ellen DeGeneres, who is also gay, said as she introduced Lil Nas X, who won two early Grammys.
Grande performed a medley of hits from her Grammy-nominated break-up album "Thank u, Next," including "7 Rings" and "Imagine."
Slain rapper Nipsey Hussle (33), who was gunned down in his Los Angeles neighborhood last year, won best rap performance for "Racks in the Middle," and was honored in a tribute by John Legend, DJ Khaled and rapper Meek Mill.
Blake Shelton and his fiancee Gwen Stefani held hands as they debuted their new romantic duet "Nobody But You," while Camila Cabello sang her recent single "First Man" to her tearful father in the audience.
Demi Lovato won a standing ovation in her first performance at a major awards show since a drugs overdose in 2018.
The Grammy winners are chosen by members of the Recording Academy, which is currently embroiled in a dispute over the departure of its new chief executive Deborah Dugan and her allegations of conflicts of interest in the nominations process.
The Recording Academy has denied the allegations.
"Wow! So many other songs deserve this," said Eilish. "I genuinely want to say I am so grateful and I am so honoured to be here. I grew up watching all of you."
Eilish and fellow fresh faces Lizzo and Lil Nas X dominated the early awards in a live show dedicated to basketball star Kobe Bryant, who died in a helicopter crash earlier on Sunday.
