Mozart in the Jungle, which stars Mexican heartthrob Gael Garcia Bernal, follows the wild personal life of a rising classical artist in a role loosely inspired by Gustavo Dudamel, the Venezuelan conductor who directs the Los Angeles Philharmonic.
"An orchestra is like a family - sometimes a dysfunctional family - of people who harness their vitality to become one big instrument," series co-creator Paul Weitz said as he accepted the award. (Also Read ? Golden Globes: Big Night for Leonardo DiCaprio and The Revenant)
The series is the latest Golden Globe for Amazon Studios, the streaming service established by the Seattle-based e-commerce giant.
Transparent, a series about a family with a transgender parent, won in the same category last year.
In other television categories, Jon Hamm won his second Golden Globe for best actor in a drama for his portrayal of womanizing advertising executive Don Draper in Mad Men, whose finale aired last year.
Mr Hamm - who also won an Emmy for the role - saluted the "incredible ride" of the show and thanked Mad Men creator Matthew Weiner, "who wrote this horrible person all the way through to the end."
Rachel Bloom won for Best Actress (Musical or comedy) series for Crazy Ex-Girlfriend.
An ecstatic Rachael hailed network The CW, a joint venture between CBS and Warner Brothers, for agreeing to put a musical comedy on network television.
"We almost didn't have a show! We made a pilot for another network and they rejected it, and we sent the pilot to every other network in Hollywood and we got six rejections in one day," she said.