President Lyndon Johnson had to rely on Everett Dirkson, the Republican leader from Illinois, to get the Civil Rights Act passed, just like Clinton had to rely on Republicans to get NAFTA passed.
That's something of a misrepresentation of facts actually, and quite a common one.
Thing is that Northern Dems and Moderate Reps were in favor of the bill, and were a few votes short to break the Southern Democrat filibuster in the Senate.
In the end, they dropped a few of the more far-reaching provisions and won the support of most Northern Conservative Republicans, and the votes to break the filibuster, as a result. That group included Dirksen, who played a large role in drafting the new, slightly watered-down version.