I'm not exactly sure of the context of the idiom you quote, but I take it to mean that you use someone else to do the morally questionable or particularly difficult deeds that you need done.
Dirty work is often used to express that some particular activity is morally questionable. To mimic your example, Uncle Sam has gotten someone else to do his dirty work. Sometimes, dirty work can be used to mean the very difficult work that is messy (sometimes literally, but figuratively could mean disorderly), but is necessary to accomplish some goal.
Bitch work or grunt work is a similar expression. It is often used to indicate the lowly work that servants would do. Things like cleaning the toilet, scrubbing the floor, etc. It does not particularly relate to your example, but I thought it was worth mentioning.
The answer suggesting "Cat's paw" reminds me of the word tool. The word tool is often used in slang terms to describe a person that is being used and doesn't even know it. See this related postSee this related post. In your example, the "someone else" might be described as a tool. Often, you will see this word used to describe a single person, not typically a nation, society, or other group.