"La La Land", a romantic musical about a struggling actress and a jazz pianist trying to make it in Hollywood, took home Globes for best comedy/musical and for its stars Ryan Gosling and Emma Stone, as well as for director Damien Chazelle, screenplay, score and original song.
"This is a film for dreamers and I think that hope and creativity are the most important things in the world," said Stone.
Independent coming-of-age film "Moonlight" was named best drama movie, while Casey Affleck, star of "Manchester by the Sea" won best drama actor for his role as a man riven by grief.
The Globe wins, handed out by the Hollywood Foreign Press Association, strengthened the front-runner status of "La La Land" and "Moonlight" for the Oscars in February.
Isabelle Huppert was named best drama actress for the French psychological thriller movie "Elle" in an upset win over Natalie Portman for "Jackie."
First-time Golden Globes host Jimmy Fallon opened the ceremony, calling it "one of the few places left where America still honors the popular vote."
Trump takes office on Jan. 20 after winning the U.S. presidential election in the Electoral College but losing the popular vote to rival Hillary Clinton.
But the evening's most dramatic moments came from Streep, the most admired actress of her generation, while accepting her lifetime achievement award.
Although the three-time Oscar winner never mentioned Trump's name, she said the most heartbreaking performance of 2016 came "when the person asking to sit in the most respected seat in our country imitated a disabled reporter."
"It kind of broke my heart when I saw it and I still can't get it out of my head because it wasn't in a movie. It was real life," Streep said.
In television, Netflix's British royal drama "The Crown" won over fantasy "Game of Thrones" and sci-fi series "Westworld" for best drama series.
Hip hop show "Atlanta" won best TV comedy series and brought Donald Glover, its creator, and star, a Golden Globe for best comedy actor.