A few minutes after he signed the Civil Rights Acton July 2, 1964, President Lyndon B. Johnson presented Hubert Humphrey, who had led the fight for its passage in the Senate, with a copy of his signing speech. On it, the president wrote, “without whom it couldn’t have happened.”

Johnson wasn’t one to share credit easily, but he understood a simple fact about Washington: Humphrey—and the dozens of other people who made the bill happen—would be relegated to a footnote, and history would give credit to the man who signed it. And he was right. Three days later, The New York Times credited Johnson as “the man who pushed [the bill] through Congress.”

This year marks the fiftieth anniversary of the Civil Rights Act, and the impression that Johnson single-handedly drove his forces in the Senate, manipulating his opponents with flawless ease, has only grown with time. In the latest volume of his acclaimed Johnson biography, 2012’s The Passage of Power, Robert Caro largely parrots Johnson’s own account of the period: “It was a struggle,” he writes, “whose strategy and day-by-day tactics were laid out and directed by him.” And the play All the Way, which opened last fall with “Breaking Bad”’s Bryan Cranston in the role of Johnson, likewise portrays the president as the omniscient political manipulator.

But this is mostly myth. Johnson had many legislative achievements during his pres­idency, but on the Civil Rights Act, he was largely ignored by his Senate allies and rebuffed by the recipients of his bear-hugging affection. The real work was performed by a long list of senators and representatives, their staffers, and a dream team of Department of Justice men who included Robert Kennedy, Nicholas Katzenbach, and Burke Marshall—not to mention civil rights leaders like Martin Luther King Jr., who built immense moral momentum behind the bill....