An article from the New York Times News service attributes Cheap Shot as originating in 1971 in a football context.

https://news.google.com/newspapers?nid=2199&dat=19810125&id=mYwyAAAAIBAJ&sjid=uucFAAAAIBAJ&pg=2377,4227702&hl=en

And rapidly making its way to political commentary soon afterwards.

(I'm on my mobile... So I hope that link resolves well)

An article from the New York Times News Service attests cheap shot as appearing in print in 1971 in a football context, with unknown origins before that.

https://news.google.com/newspapers?id=mYwyAAAAIBAJ&sjid=uucFAAAAIBAJ&pg=2377%2C4227702

And rapidly making its way to political commentary soon afterwards. If we can trust the research of a random columnist from 1981 on the early 70's origin, it would make sense for it to grow quickly to common use, since both televised football and politics experienced a boom in that era.

The cheapness is in the sense of "deserving of contempt" (similar to the phrase "a cheap trick").