Install mysql-client if not installed, this will provide the mysqldump utility

apt-get install mysql-client
cp /usr/share/doc/rsnapshot/examples/utils/backup_mysql.sh /usr/local/bin/

Make sure your backup scripts are owned by root, and not writable by anyone else.

chown root.root /usr/local/bin/backup_mysql.sh
chmod o-w /usr/local/bin/backup_mysql.sh

This scripts is designed only to back up all databases to a single file.

Edit script

nano /usr/local/bin/backup_mysql.sh

replace

/usr/bin/mysqldump --all-databases > mysqldump_all_databases.sql

with

/usr/bin/mysqldump --defaults-file=/etc/mysql/debian.cnf --all-databases > mysqldump_all_databases.sql

If you need to dump a single database use line below

/usr/bin/mysqldump --defaults-file=/etc/mysql/debian.cnf DATABASE > DATABASE.SQL

Add script to rsnapshot

Edit rsnapshot config

nano /etc/rsnapshot.conf

Under BACKUP POINTS / SCRIPTS add following

backup_script /usr/local/bin/backup_mysql.sh localhost/mysqldump

Test config

rsnapshot configtest