Plesk MySQL server have no root user. Instead Plesk have user “admin” with root privileges.

mysql -u admin -p`cat /etc/psa/.psa.shadow`

To find password for user admin, run

[root@server22 ~]# cat /etc/psa/.psa.shadow
FfRBehWy
[root@server22 ~]#

To login to MySQL

mysql -u admin -pFfRBehWy

For example

[root@server22 ~]# mysql -u admin -pFfRBehWy
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 2555 to server version: 4.1.20

Type 'help;' or '\h' for help. Type '\c' to clear the buffer.

mysql>

To login with out password, create a file /root/.my.cnf

vi /root/.my.cnf

With following content

[mysql]
user = admin
password = FfRBehWy

[mysqldump]
user = admin
password = FfRBehWy

This will allow you to use mysql and mysqldump with out providing passwords.