It's actually an acl issue and not a permissions issue as you would first suspect. You have to set the acl option on the file system hosting your web server, and do it in the /etc/fstab file so it will survive reboots. The line in my /etc/fstab looks like:
UUID=397966d0-61eb-4ef4-8ffd-10b6c06964dd / ext4 defaults,acl 1 1
Next, remount the file system as follows for acl's to take effect:
mount -o remount /
Next, set an acl on the parent directory for uploads. In my case, it's as follows:
setfacl -R -m u:apache:rwx /var/www/html
Next, check to make sure that 'apache' has rwx access to the directory:
getfacl /var/www/html
You will see an entry displaying that 'apache' now has access for that directory, and it will look like the following in the output:
user:apache:rwx
Now, you can go into Wordpress, and all should be well.