The 73-year-old died on Wednesday in New York from complications from cancer.
Demme remains best known for the 1991 horror-thriller, starring Anthony Hopkins as serial killer Hannibal Lecter and Jodie Foster as FBI agent Clarice Starling.
The movie won five Academy Awards, including best picture, best actor for Hopkins and best actress for Foster.
Demme went on to direct "Philadelphia" in 1993, a ground-breaking movie that won Hanks his first Academy Award for portraying a lawyer fired by his firm for contracting HIV and fighting for justice.
"I can confirm that Jonathan passed away early this morning in his Manhattan apartment, surrounded by his wife, Joanne Howard, and three children," his publicist said in a brief statement.
Demme died from complications from esophageal cancer and is survived by his three children, Jos, 21, Brooklyn, 26, Ramona, 29 and her husband James Molloy.
His death sparked an immediate outpouring of grief from fans, friends and former colleagues for a director who was widely respected in Hollywood and beyond, feted also for his work on behalf of refugees.
Director Barry Jenkins whose film "Moonlight" won best picture at this year's Academy Awards wrote: "Met tons through the Moonlight run but my man Demme was the kindest, most generous. A MASSIVE soul. He lived in love. And rests in peace."
In all, Demme directed 20 movies and 12 documentaries, among a raft of other credits. Neither was he afraid to shy away from America's tortured past.
In 1998 he directed Oprah Winfrey in the movie "Beloved" based on the Toni Morrison novel about a slave visited by the spirit of her dead daughter.
Unveiling his movie "Rachel Getting Married" at the Venice film festival in 2008, two months before Barack Obama was elected America's first black president, Demme paid tribute to a multicultural United States.
Asked about the movie's wedding party -- with a white bride (Rosemarie Dewitt), black groom (Tunde Adebimpe) and women who wear saris -- he said at the time that it was a "real picture of America today."
"That's the America I love," Demme said, adding that it reflected "the America that I feel very deeply connected to."
In place of flowers, Demme's family has requested donations to be made to a Florida-based charity that works to protect immigrants, Americans For Immigrant Justice.
Republican President Donald Trump has appalled liberals and activists across the country by seeking to restrict access for immigrants and refugees, and by pledging during last year's campaign to deport millions of undocumented workers.
In 2013, the Florida charity announced that it was honoring Demme with a human rights award for "his tireless work over the decades on behalf of Haitian refugees and other vulnerable immigrants, through his advocacy and his art."
Among Demme's documentaries was "Man from Plains," about the life of former Democratic president Jimmy Carter, which followed Carter on a promotional tour for his book "Palestine Peace, not Apartheid" and touched on his post-presidential life.
"Deeply saddened by the passing of the most brilliant man -- director, father, friend, activist," wrote British actress Thandie Newton on Twitter. "Devastating to let him go. I love you," she added.
Thierry Fremaux, director of the Cannes film festival in France, also paid tribute. "A wonderful man. Sadness," he tweeted.
English rocker Robyn Hitchcock, whose music was so embraced by Demme that he filmed a concert movie of him in 1998, was also among the mourners.
"He loved people, he loved filming them; I am so sad to hear he's gone," Hitchcock wrote on Twitter.
Demme will be laid to rest at a private family funeral.