It is not rare situation that I have mysqldump file which contains more than one database. Here is example how you can extract and import just one database from mysqldump file. In this example I will use database zezalica and root password is pillow.
One of possible solutions is to directly import database from mysqldump file.
Also you can create new sql file with just one database backup.
Now you should import that dump file. Just one small note, zezalica.sql should be owned by user mysql and group mysql, so I copy that file in /var/lib/mysql and change ownership according that.
mysql> use zezalica;
mysql> source /var/lib/mysql/zezalica.sql
Simple as that!