Here are simple steps to update MySQL database server 5.1 to MySQL database server 5.6 for Centos web panel:
Login to your server with root username and password from SSH:
1. Create full database backup (In case mysql server break we have backup to restore).
mysqldump --events --all-databases | gzip > /home/alldatabases.sql.gz
2. Install Remi repo
cd /etc/yum.repos.d wget http://rpms.famillecollet.com/enterprise/remi.repo
3. Now upgrade MySQL Server.
yum --enablerepo=remi update mysql-server sudo mysql_upgrade
4. Restart MySQL Server.
service mysqld restart
That is all.