You need to setup your web application to use claims authentication and then implement a custom membership provider. There is some implementations for mysql membership providers out there:
http://www.codeproject.com/Articles/12301/Membership-and-Role-providers-for-MySQL
Here is an article about how to setup claims authentication with an ASP.NET membership provider:
http://msdn.microsoft.com/en-us/library/gg252020(v=office.14).aspx
Obviously you need to adapt the implementation to the MySQL Membership provider which should be easy enough.
Sorry about providing links but the ammount of steps is so big that is not suitable to be posted here.