Leyva was born in Baracoa, Cuba, and raised in the Bronx, New York City. She is of Cuban and Dominican descent and has appeared in a number of Off-Broadway productions.

On television, Leyva had a recurring role of Detective Mariluz Rivera in Law & Order and also appeared in different roles on the show and on Law & Order: Special Victims Unit. She has guest-starred on Third Watch, The Sopranos, The Good Wife, Girls, and Elementary.

In 2013, Leyva began starring as Gloria Mendoza, a Latina leader in the jail, in the Netflix comedy-drama series Orange Is the New Black. Her character was recurring for the first two seasons and she was promoted to series regular in Season 3.

In 2015, she co-starred opposite Viola Davis and Catalina Sandino Moreno in the courtroom drama film Custody directed and written by James Lapine.