I Can't Breathe, by the R&B singer H.E.R. was released last summer, and quotes Floyd's last words, as he died in police custody in Minneapolis.
Accepting the prize, the star said she never imagined "that my fear and my pain would turn into impact".
"That's why I write music," she added, "and I'm so, so grateful"
The singer, whose real name is Gabriella Wilson, was accompanied by her co-writer Tiara Thomas, and she recalled how they'd written I Can't Breathe "over FaceTime" last summer.