The idea is, miserable though the conditions in a sweatshop may be, they are better than the alternatives the workers face, or else they wouldn't choose to be in the sweatshop. If donating $0 to that country is permissible, then surely doing something that makes things slightly better is at least as permissible.

Also, if the company that runs the sweatshop is in fact making huge profit margins from it (I assume that's one thing "exploitation" implies), and if the company isn't especially unusual, then that suggests there's a lot of room for some other company to set up another factory, offer a slightly higher wage or better conditions to attract workers away from the first one, and still make great profits. If there are no artificial barriers to this, then foreign companies competing for workers may end up really improving matters after a while. Contrariwise, if there are significant risks to setting up more sweatshops—e.g. suffering bad PR, running afoul of new or newly interpreted legislation—then the few sweatshops will tend to remain few and miserable, and be run by companies that are more immune to the above dangers (probably large, with good legal departments and/or political pull, and those that don't depend as much on a good reputation).

This implies that people who manage to cause serious problems for companies by protesting their use of sweatshops are probably making things worse for the workers. It is interesting to reflect on.