While it is true that without the Republicans the 1964 Civil Rights Act would not have passed, it wouldn't have passed without the Democrats that supported it either. A majority of Democrats did support it, as did most Republicans. What is interesting is the Democratic split which was eerily reminiscent of the 1860 election. The following is the breakdown from the Wikipedia entry for the Civil Rights Act of 1964, footnoted and consistent with the account in James T. Patterson's, "Grand Expections: The United States from 1945-1974," (Oxford University Press 1997):

"Vote totals
Totals are in "Yea-Nay" format:

The original House version: 290-130 (69%-31%)
The Senate version: 73-27 (73%-27%)
The Senate version, as voted on by the House: 289-126 (70%-30%)

By party

The original House version:

Democratic Party: 152-96 (61%-39%)
Republican Party: 138-34 (80%-20%)

The Senate version:

Democratic Party: 46-21 (69%-31%)
Republican Party: 27-6 (82%-18%)
The Senate version, voted on by the House:[9]

Democratic Party: 153-91 (63%-37%)
Republican Party: 136-35 (80%-20%)

By party and region
Note: "Southern", as used in this section, refers to members of Congress from the eleven states that made up the Confederate States of America in the American Civil War. "Northern" refers to members from the other 39 states, regardless of the geographic location of those states.

The original House version:

Southern Democrats: 7-87 (7%-93%)
Southern Republicans: 0-10 (0%-100%)
Northern Democrats: 145-9 (94%-6%)
Northern Republicans: 138-24 (85%-15%)

The Senate version:

Southern Democrats: 1-20 (5%-95%) (only Senator Ralph Yarborough of Texas voted in favor)
Southern Republicans: 0-1 (0%-100%) (this was Senator John Tower of Texas)
Northern Democrats: 45-1 (98%-2%) (only Senator Robert Byrd of West Virginia opposed the measure)
Northern Republicans: 27-5 (84%-16%) (Senators Bourke Hickenlooper of Iowa, Barry Goldwater of Arizona, Edwin L. Mechem of New Mexico, Milward L. Simpson of Wyoming, and Norris H. Cotton of New Hampshire opposed the measure)"