The actress plays Annalise Keating - a brilliant but conflicted criminal defence professor who, with five of her students, becomes entwined in a murder plot - in How to Get Away With Murder.
Her win came after two other black women took home Emmy awards - Regina King for American Crime, and Uzo Aduba for Orange Is the New Black.
Here is a transcript of her speech.
"'In my mind, I see a line. And over that line, I see green fields and lovely flowers and beautiful white women with their arms stretched out to me over that line. But I can't seem to get there no how. I can't seem to get over that line.'
"That was Harriet Tubman in the 1800s. And let me tell you something: The only thing that separates women of colour from anyone else is opportunity. You cannot win an Emmy for roles that are simply not there.
"So, here's to all the writers, the awesome people that are Ben Sherwood, Paul Lee, Peter Nowalk, Shonda Rhimes. People who have redefined what it means to be beautiful, to be sexy, to be a leading woman, to be black.
"And to the Taraji P Hensons and Kerry Washingtons, the Halle Berrys, the Nicole Beharies, the Meagan Goodes, to Gabrielle Union. Thank you for taking us over that line. Thank you for the Television Academy. Thank you."
Actress Viola Davis received a standing ovation for her acceptance speech at the Emmy Awards, where she became the first black woman to win outstanding actress in a drama.