Play doesn’t come with out-of-the-box support for Flyway, however Toshiyuki Takahashi has created a Play plugin to substitute Evolutions for Flyway.
This lets you run your Flyway migrations on Play application startup.
For Play 2.1, 2.2 and 2.3 you can use the play-flyway plugin.
For Play 2.4 and newer you can use the flyway-play module.
Both sites contain instructions on how to get started quickly as well as comprehensive documentation on usage and configuration.