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."

Mourad Laachraoui, brother of Najim Laachraoui, won the -54kg final in Montreux, Switzerland, on Thursday.
He beat 18-year-old Spanish fighter Jesus Tortosa 6-3 to claim the title.
The 21-year-old, who will compete at Rio 2016, has said he was "scared and saddened" by his brother's role in the attack on the Belgian capital in March.
Speaking at a news conference he arranged in Brussels following the attacks, Mourad Laachraoui also said no-one in his family had heard from his brother since 2013.
Najim, 24, was one of two men who carried out devastating bomb attacks in the check-in area of Zaventem airport on 22 March. A third bomb was detonated at Maelbeek metro station shortly after, with a total of 32 people killed.
Prosecutors have also linked him to November's attacks in Paris in which 130 people died.
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.
[NEXT_CONCEPT]
A Belgian whose older brother was one of the suicide bombers in the terror attacks on Brussels has won gold at the European Taekwondo Championships.