The San Francisco Chronicle said Mr Baker died in his sleep at his home in New York on Thursday night.
He initially designed an eight-colour flag in 1978 for the city's gay freedom day, the precursor to the modern pride parade.
A candlelight vigil is planned for Friday evening in San Francisco.
Screenwriter Dustin Lance Black tweeted: "Rainbows weep. Our world is far less colourful without you, my love. Gilbert Baker gave us the rainbow flag to unite us. Unite again."
California State Senator Scott Weiner said Mr Baker's work "helped define the modern LGBT movement".
"Rest in power, Gilbert," he said.
Mr Baker's original flag had eight colours, each representing a different aspect of humanity:
§ Pink - sexuality
§ Red - life
§ Orange - healing
§ Yellow- sunlight
§ Green - nature
§ Turquoise - art
§ Indigo - harmony
§ Violet - human spirit
It would later be reduced to six stripes, removing pink and indigo, and swapping blue for turquoise.
Mr Baker said he wanted to convey the idea of diversity and inclusion, using "something from nature to represent that our sexuality is a human right".
In 2015, New York's Museum of Modern Art acquired the flag for its design collection, calling it a "powerful design milestone".
"I decided that we should have a flag, that a flag fit us as a symbol, that we are a people, a tribe if you will," Mr Baker told the Museum in an interview.
"And flags are about proclaiming power, so it's very appropriate."
A giant rainbow flag flies at the junction of Castro and Market streets in San Francisco, near Harvey Milk Plaza, which is named for America's first openly gay politician.
A candlelit vigil is planned for 19:00 local time (02:00 GMT) beneath the flag.