There was a documentary about LBJ on PBS a while ago showing all the calls and favors he made to get the votes needed to pass, and all the arm twisting he did.
Passage in the House of Representatives
The bill was sent to the House of Representatives, and referred to the House Judiciary Committee, chaired by Emmanuel Celler. After a series of hearings on the bill, Celler's committee greatly strengthened the act, adding provisions to ban racial discrimination in employment, providing greater protection to black voters, eliminating segregation in all publicly-owned facilities (not just schools), and strengthening the anti-segregation clauses regarding public facilities such as lunch counters. They also added authorization for the Attorney General to file lawsuits to protect individuals against the deprivation of any rights secured by the Constitution or U.S. law. In essence, this was the controversial "Title III" that had been removed from the 1957 and 1960 Acts. Civil rights organizations pressed hard for this provision because it could be used to protect peaceful protesters and black voters from police brutality and suppression of free speech rights.
The bill was reported out of the Judiciary Committee in November 1963, and was then referred to the Rules Committee, whose chairman, Howard W. Smith, a Democrat from Virginia, indicated his intention to keep the bill bottled up indefinitely. It was at this point that President Kennedy was assassinated. The new president, Lyndon Johnson, utilized his experience in legislative politics and the bully pulpit he wielded as president in support of the bill.
Because of Smith's stalling of the bill in the Rules Committee, Celler filed a petition to discharge the bill from the Committee. Only if a majority of members signed the discharge petition would the bill move directly to the House floor without consideration by Smith's committee. Initially Celler had a difficult time acquiring the signatures necessary, as even many congressmen who supported the civil rights bill itself were cautious about violating House procedure with the discharge petition. By the time of the 1963 winter recess, 50 signatures were still needed.
On the return from the winter recess, however, matters took a significant turn. The pressure of the civil rights movement, the March on Washington, and the President's public advocacy of the Act had made a difference of opinion in Representatives' home districts, and soon it became apparent that the petition would acquire the necessary signatures. To prevent the humiliation of the success of the petition, Chairman Smith allowed the bill to pass through the Rules Committee. The bill was brought to a vote in the House on February 10, 1964, and passed by a vote of 290 to 130, and sent to the Senate.