AutoRip, which is only available in the US, will automatically keep a digital copy of eligible CDs in a customer's cloud storage account.
Customers will be able to access the music via Amazon's Cloud Player on the web or via tablet and smartphone apps.
Amazon has drawn up a catalogue of 50,000 CDs that are eligible for AutoRip.
The catalogue has been compiled from those albums that have proved most popular with Amazon customers in the last 15 years.
The list includes "Dark Side of the Moon" by Pink Floyd, "Thriller" by Michael Jackson and "21" by Adele.
Any customer who has bought a CD in the catalogue from Amazon since the firm started trading in 1998 will be eligible to get a free MP3 copy of it. Amazon said it anticipated creating copies of millions of CDs.
"When we picked those 50,000 titles we focused on having a substantial majority of our physical CD sales covered," said Steve Boom, head of digital music at Amazon in a statement.
The service potentially makes it much easier for people to build up a library of digital music. Before now most CD owners had to rip the songs themselves to create digital versions.
The Amazon Cloud Player can be accessed a web browser, as well as on Android phones, iPhones, Kindle Fire tablets and other devices.
The move is widely seen as an attempt to take market share from arch-rival Apple's iTunes music store.
The Cupertino giant has similar cloud storage services for music and leads the market with more than 50% of the market share, while Amazon has less than 15%. Google offers a similar service.

Rat droppings and a live mouse were found in the kitchen at Shimla Pinks, in Leicester, on Tuesday.
Officers shut the restaurant while people were dining, with the business fined Â£1,000 during a hearing at the city's magistrates.
Owner Meena Patel told the Leicester Mercury he felt "terrible" about the problem and vowed to rectify it.
Leicester City Council health inspectors confirmed the closure during the hearing on Thursday.
The court was told droppings were found behind the freezer and there were holes in doors and the skirting boards, allowing the rodents access to food preparation areas.
Bacterial infections such as Leptospirosis and Ecoli can be spread by urine from mice and rats.
The council said in a statement the owners are not allowed to open until it was satisfied they have "addressed the problems".
The BBC approached Shimla Pinks for comment but has yet to receive a reply.
However, Mr Patel told the Mercury he was going to "eradicate" the problem.
The premises, in London Road, will close for a week to allow walls to be rebuilt and kitchen equipment replaced.
Shimla Pinks has earned consistent positive reviews on Trip Advisor, earning it a "certificate of excellence" in 2015.
Online retailer Amazon has launched a service that stores free digital versions of CDs bought via its store.
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A curry house has been shut down after an infestation of rats and mice was found by health inspectors.