The major connection here is that the needs of the Cold War helped cause some Americans to support civil rights for African Americans. A really good example of this can be seen in this campaign ad from Richard Nixon in his presidential campaign of 1960.
There, Nixon tells us that civil rights are important for Cold War purposes. He says that we need to use the talents of all our citizens to compete against communism. He also says that when we fail to grant civil rights to some in our society it makes us look bad in the eyes of the world. What he means is that segregation and discrimination make us look bad to the people (non-white, of course) of Africa and Asia. We need to look good to them so they will support us against the Soviets.
In these ways, the needs of the Cold War helped to bring about some amount of white support for civil rights.