Clearly, the civil rights movement won their battle to end segregation. However, I would not say that the movement really undermined the culture of Jim Crow.
Instead, I would say that the movement got enough support outside the South to push through laws that ended segregation even though the South still wanted it.
There were lots of outside elements that contributed. The two main ones I would mention are
- Movement of blacks to the North in WWII. There, they could vote and they became an important voting bloc in some parts of the North.
- Cold War -- made the US want to look good in the eyes of the world and that meant that high government officials thought it was in our best interests to end segregation so that black and brown people around the world would like us.