It is expected to be rolled out in the UK, Germany and Australia later this year, although Amazon has not yet released a definite timeline.
The service will cost $3.99 (£3.25) a month for owners of its Amazon Echo speaker and $7.99 (£6.52) a month for Amazon Prime members.
Everyone else will have to pay $9.99 (£8.15) a month, the same charge as US subscribers to Spotify and Apple music.
Amazon will continue to offer Prime members a limited streaming service for free.
The tech giant is pinning a lot on the Echo system, a smart speaker that responds to voice commands.
Steve Boom, vice president of Amazon Music, said: "The first phase of growth (in music streaming) was driven almost entirely by smartphones.
"We believe pretty strongly that the next phase of growth in streaming is going to come from the home."
The launch comes as the way people listen to music is changing.
During the first half of 2016, digital purchasing of single tracks was down 24% and digital albums were down 18%, while streaming was up 59% compared with the same period a year ago.
Amazon has also worked Artificial Intelligence into its Echo system, so users can request songs that fit a particular mood or can search for a song via lyrics.
Amazon Echo works through the Alexis voice system, which has been refined for the new music service.
"Even something like playing a new song, that`s not easy," said Mr Boom. "If you say, `play Adele`s new single,` it`s now the third single off the album.
"All the songs have the exact same release date, so we had to train Alexa to get smarter about looking at songs that are rising the charts and are the newest songs being played on the radio."
Kintan Brahmbhatt, director of Amazon Music, said: "You can ask for Michael Jackson by saying, `Play music by the King of Pop.`
"It`s smart enough to know that`s what you meant."
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