Context: It was common that software vendors sell licenses to their clients to use software products, such as Enterprise Resource Planning, which are deployed as a monolithic entity on clients' premises. Moreover, many clients, especially big organizations, often require software products to be customized for their specific needs before deployment on premises. Objective: However, as software vendors are migrating their monolithic software products to Cloud-native Software-as-a-Service (SaaS), they face two big challenges that this paper aims at addressing: (1) How to migrate their exclusive monoliths to multi-tenant Cloud-native SaaS; and (2) How to enable tenant-specific customizations for multi-tenant Cloud-native SaaS. Method: This paper suggests an approach for migrating monoliths to microservice-based Cloud-native SaaS, providing customers with a flexible customization opportunity, while taking advantage of the economies of scale that the Cloud and multi-tenancy provide. We develop two proofs-of-concept to demonstrate our approach on migrating a reference application of Microsoft called SportStore to a customizable SaaS as well as customizing another Microsoft's microservices reference application called eShopOnContainers. Results: We have shown not only the migration to microservices but also how to introduce the necessary infrastructure to support the new services and enable tenant-specific customization. Conclusions: Our customization-driven migration approach can guide a monolith to become SaaS having (synchronous and asynchronous) customization power for multi-tenant SaaS. Furthermore, our event-based customization approach can reduce the number of API calls to the main product while enabling different tenant-specific customization services for real-world scenarios.