The awards were presented in a star-studded ceremony Monday evening at Carnegie Hall.
Hillary Clinton joined daughter Chelsea for her first evening out since the birth of daughter Charlotte Clinton Mezvinsky on Sept. 26.
"I had sort of wondered if I'd get to wear a fancy dress and high-heeled shoes again," Clinton said, drawing laughter and applause from the crowd.
Clinton was honoured for her work with the Bill, Hillary & Chelsea Clinton Foundation. Glamour also cited her work on a partnership between the Clinton Foundation and the Bill & Melinda Gates Foundation to encourage achievement among women and girls across the world. That commitment, Clinton said, was only strengthened by becoming a mother.
"I want Charlotte to grow up to be whatever she dreams to be, wherever she dreams to be it," Clinton said.
Nyong'o, whose mother offered congratulations on a pre-taped video, won a supporting-actress Oscar for her performance in "Twelve Years a Slave," capping her swift rise to the top rungs of Hollywood and the fashion world. She was also named a new face of Lancome and appeared on the cover of People magazine's Most Beautiful issue. She will be appearing in the new "Star Wars" movie.
"The truth is, I don't really know what's next," Nyong'o said, through tears. "Though what has happened to me so far is incredible, I am still a work in progress and I hope to always be."
Cox, from "Orange Is the New Black," was the first openly transgender person to be nominated for an Emmy Award. In a press release, Glamour called her "the face of one of the biggest equality stories of 2014."
"I love being a woman, and I am so honoured to be the vessel tonight for what I believe is this inevitable historic moment where the womanhood of transgender women is not in question," Cox said.
Burton is best known for designing Kate Middleton's wedding gown. She became creative director of Alexander McQueen in 2010, following the designer's death, and has been credited, Glamour noted, with boosting the British fashion industry.
Power, the U.S. ambassador to the United Nations, was cited for her work in diplomacy "during a year when the U.N. Security Council held more emergency sessions than it had in the entire 10 years prior."
Other Glamour winners:
—Robin Roberts, the "Good Morning America" host who has survived breast cancer and a public struggle with myelodysplastic syndrome, a rare blood and bone marrow disease, cited for her success and influence in broadcasting.
—Actress Mindy Kaling, creator and star of the Fox comedy "The Mindy Project." ''Millions of women feel like Kaling is their BFF," Glamour said.
—Natalia Vodianova, supermodel and founder of the Naked Heart Foundation, which helps children with special needs.
—Sylvia Earle, a noted oceanographer who "even as her eightieth birthday approaches ... continues to spend three months a year on expeditions."