A streamlined approach enables customers to retain management control over their data in a database-as-a-service (“DBaaS”) setting, by providing managed backup copies outside cloud service providers' sphere of control. An illustrative data storage management system provides control over performing backup operations to generate managed backup copies, storing managed backup copies, recovering managed backup copies in whole or in part, migrating managed backup copies, and migrating DBaaS instances. Management control also extends to choices of where to store the managed backup copies, whether on the same cloud computing platform as the source DBaaS, on a different cloud computing platform, and/or in a non-cloud data center.