This issue can also be a result of having SELinux enabled. This can be solved using:
chown -R apache:apache /var/www/html/directory_to_write
chcon -R -t httpd_sys_content_t /var/www/html/directory_to_write
chcon -R -t httpd_sys_rw_content_t /var/www/html/directory_to_write
You can find more about the contexts at https://access.redhat.com/documentation/en-US/Red_Hat_Enterprise_Linux/6/html/Managing_Confined_Services/sect-Managing_Confined_Services-The_Apache_HTTP_Server-Types.html
