Davis completed her collection by winning best audio book for her autobiography Finding Me.
"I wrote this book to honour the six-year-old Viola," said the star. "To honour her life, her joy, her trauma, everything."
The star won the best supporting actress Oscar in 2016 for Fences.
Her Emmy Award recognised the TV drama How to Get Away with Murder, and she has two Tony Awards for her theatre work - featured actress in a play for King Hedley II (2001) and lead actress in a play for Fences (2010).
"I just EGOT!" announced the star at the Grammts, becoming visibly emotional as she thanked her family for being "the best chapter in my book".
The 17 other EGOT winners include Sir John Gielgud, Rita Moreno, Andrew Lloyd Webber, John Legend and Jennifer Hudson.
Davis picked up her prize at the Grammys "premiere ceremony", which mostly recognises technical and genre categories.
Beyoncé has already won two prizes: Best dance recording for the song Break My Soul, and traditional R&B performance for Plastic Off The Sofa.
The awards bring her one step closer to becoming the most-awarded artist in Grammy history.
The star now has 30 trophies, just one short of equalling the current record holder, Hungarian-British conductor Georg Solti.
Harry Styles also picked up best engineered album for Harry's House, which is also up for the night's biggest prize - album of the year.
But the British star faces tough competition for that award, with Adele, Beyoncé and Latin star Bad Bunny among the other favourites.