"When a SqlConnection is successfully opened, the failover partner name is returned by the server and supersedes any values supplied in the connection string” indicates the failover partner name is returned. This would be the same failover partner name that you use for the Primary/Mirror relationship.
It sounds like in your case you would want to make sure that the application is able to contact the mirror based upon that failover partner name. If you run the query on the primary in the mirror:
select * from sys.database_mirroring
You can see the mirroring_partner_instance will be what is returned and what it is going to connect to. I hope this information helps.