TV programmes, comedy series and films such as The Wolf of Wall Street are available via the service that already lets people stream video content.
Amazon said it would not levy any additional charge for the service. In the UK, a year’s subscription to Prime costs £79.
The offline option is available in the US, UK, Germany and Austria.
Owners of Amazon’s Fire tablet who are also signed up to the Prime service have been able to download movies and programmes for months. An updated Amazon Video app now extends this to non-Amazon branded gadgets.
Amazon is the first net-based video subscription service to offer the downloading option on Android and iOS devices.
Licensing restrictions will mean some content will be available only for streaming rather than downloading.
In a statement, Amazon said its own video series were all available offline and added a list of many other popular programmes, including 24, Downton Abbey, Hannibal and Vikings, that were also available.