It is not unusual for bin file to fill up the partition for mysql when one is doing master-slave replication.
Ideally one should have a reasonable setting in the my.cnf file. Setting the expire_logs_days to something reasonable or the max_binlog_size to a size you one can leave with.
In the case where one missed the opportunity to do so at the setup time and one cannot afford to change this and do a restart. two commands come in handy. One will reduce to the file desired and the other will reduce the bin files to the desired date.
PURGE BINARY LOGS BEFORE '2015-08-03 23:00:00';
PURGE BINARY LOGS TO 'mysql-bin.03';
These work immediately and results should show of course one case just use now() or something similar in the first case as opposed to typing in a date.
As an example, you can purge bin logs older than three(3) days using the following
PURGE BINARY LOGS BEFORE now() - interval 3 day;