https://www.senate.gov/artandhistory/history/civil_rights/cloture_finalpassage.htm

To pass a civil rights bill in 1964, the Senate proponents of that bill developed a three-part strategy. First, Majority Leader Mike Mansfield maneuvered the bill away from the Judiciary Committee and made it the Senate’s pending business. Second, a bipartisan legislative team of senators and staff, led by Majority Whip Hubert Humphrey and Minority Whip Thomas Kuchel, developed a plan to defeat a well-organized filibuster. Finally, they enlisted the aid of Minority Leader Everett Dirksen. Only Dirksen could provide the Republican votes needed to invoke cloture and bring about passage of the bill. “The bill can’t pass unless you get Ev Dirksen,” President Lyndon Johnson told Hubert Humphrey. “You get in there to see Dirksen. You drink with Dirksen! You talk with Dirksen. You listen to Dirksen.”

In an era when there were many factional divisions within both political parties, the biggest headaches for Democratic leader Mike Mansfield often came not from Republicans but from the conservative bloc of his own party caucus. The filibuster that threatened to derail the civil rights bill in 1964 was not led by the opposition party, but by an opposing faction within the majority party. To invoke cloture on the civil rights bill, Democratic proponents of the bill needed strong Republican support. If the bipartisan team could gain the support of Dirksen, a small-government conservative from Illinois, they might win over other conservatives.

This presented Everett Dirksen with a dilemma. It was a presidential election year and, as one historian commented, Dirksen was asked “to deliver Republican votes in support of a Democratic president who could not bring along enough of his own party to seal the deal.” As the long civil rights debate unfolded, it did so with the backdrop of presidential primaries. The last thing the Senate’s Republican leader should be doing, many argued, was to provide the Democratic administration with a major legislative victory, but Dirksen, a proud Republican from the Land of Lincoln, was determined to preserve the Republican legacy inherited from the Great Emancipator. In addition, there were liberal and moderate Republicans who were deeply committed to the cause of civil rights, and senators such as Jacob Javits of New York urged Dirksen to take immediate action. On the other hand, staunch conservatives like Bourke Hickenlooper of Iowa fought Dirksen every step of the way. Dominating the GOP caucus, many conservatives believed the civil rights bill represented an unprecedented intrusion by the state into the daily lives of Americans.

By late February 1964, as the bipartisan team set to work, Dirksen began tinkering with the bill. Over the next three months, the Republican leader, meeting daily with Humphrey or Kuchel but largely avoiding his caucus, suggested a host of amendments divided into categories of technical and substantive. The lesser amendments corrected or clarified language, while substantive amendments brought compromise among competing views. Throughout the negotiations, Dirksen kept his own counsel. “What is Ev Dirksen up to?” asked the Los Angeles Times. Dirksen is “the master of obscure intention,” wrote the Washington Post, which will be “revealed only in his own good time.” While Dirksen worked with the bipartisan team, key staff negotiated with individual Republican senators.

In early April Dirksen attended the Republicans’ weekly policy luncheon and presented a set of 40 draft amendments. Conservatives, suspicious of the leader’s behind-the-scenes deal-making, expressed only reluctant support. The liberals simply rebelled, accusing Dirksen of watering down the House-passed bill. As the meeting broke up, it was clear that the Republican caucus remained divided. Reassuringly, Senator Humphrey expressed optimism. Dirksen’s “not trying to be destructive,” Humphrey commented. “He’s trying to be constructive.”

The debate in the Senate Chamber continued, as Dirksen produced more amendments while constantly testing the waters looking for support. Details were discussed, agreements were made, and deals were struck as Dirksen worked to gain votes for cloture while maintaining the integrity of the House-passed bill. As Kentucky senator John Sherman Cooper saw it, Dirksen’s proposal would not weaken the bill, but would be “a substantial amendment in developing sentiment for the bill, not only here, but throughout the country. It is going to have appeal.”

https://www.archives.gov/publications/prologue/2004/summer/civil-rights-act

Lawrence F. O'Brien, President Kennedy's and later President Johnson's chief of liaison with the Congress, recalled it this way:

[Y]ou had a battle on two fronts simultaneously. You had a battle with the conservatives on the committee, the southern Democrats, conservative Republicans, but you had just as tough a battle with the liberals. Their position was the old story of the half loaf or three-quarters of a loaf, and [now they were saying] "we'll settle for nothing less [than the whole loaf.]" . . . We shared their views, and we'd love to do it their way.

We were accused by some of being weak-kneed but, my God, are you going to have meaningful legislation or are you going to sit around for another five or ten years while you play this game? Those liberals sat around saying, "No, we won't accept anything but the strongest possible civil rights bill, and we won't vote for anything less than that." To kill civil rights in that Judiciary Committee was an appalling possibility! And it was not only a possibility, it came darn close to an actuality.

Curse those liberals for demanding a stronger civil rights bill, right!?

Ergo: Liberals supported; conservatives resisted. No tankies in Congress. Thank a liberal. Yes, I'm aware that what is progressive for the time is comparatively conservative by today's standards; that doesn't change the point.