The internet retailer launched its Cloud Player in the US, ahead of rivals Apple and Google which are rumoured to be developing similar systems.
Users are given 5Gb of storage space, roughly equivalent to 1,200 tracks, but can opt to pay for additional capacity.
The Cloud Player is currently only available through web browsers and mobile devices running Google Android.
Commenting on the launch, Amazon's vice president of movies and music, Bill Carr said: "Our customers have told us they don't want to download music to their work computers or phones because they find it hard to move music around to different devices."
Although a number of smaller cloud music services already exist, such as mSpot and AudioBox, Amazon is the first of the big technology companies to venture into this area.
Speculation has been rife that Apple would launch a cloud based version of iTunes since it purchased the online music service Lala in December 2009.
It is widely expected that Apple's offering will form part of a broader re-launch of the MobileMe platform.
Google, which already offers cloud services in the form of Gmail and Google Documents, is also believed to be testing a music storage system, or "locker".
It is not known what agreement, if any, Amazon has reached with the four major record companies, regarding users uploading copies of their music.
Making online copies of tracks is known as format shifting. While the practice may violate copyright, in the US, it is generally defensible under the principle of fair usage.
The same rules do not apply in the UK - meaning, for example, it is technically a breach of copyright law to copy music from a CD onto an MP3 player.
However the music industry has generally turned a blind eye to users copying legally purchased music, not least because of the difficulty in policing infringement.
If Amazon Cloud is to launch in the UK, the company may have to address those issues, say lawyers.
"I am guessing that what they are doing in the US is using the fair usage laws that cover format shifting," said Brett Farrell, a technology and media lawyer at Barlow Robbins.
"Technically you do not have the right to format shift in the UK.
"If a major player moved into town and wanted to encourage format shifting then I think the record companies would use that as a way of getting them to the negotiating table," he said.