In his first public acknowledgment, Washington apologized for his behavior, eight months after he told Rangers president Nolan Ryan, who turned down the manager's offer to resign.
"I made a huge mistake and it almost caused me to lose everything I have worked for all of my life," Washington said at a news conference. "I am not here to make excuses. There are none."
Washington, 57, said he used cocaine only once and called it "stupid" and "shameful."
SI.com first reported the failed test. Washington said he told the commissioner's office and Rangers officials about using cocaine before he had a routine drug test.
"He came forward and said he would resign," Ryan said. "He understood the consequences. We had a lot of discussions and a lot of soul-searching on it.
"We felt like he was sincere and forthright. We are very disappointed by this. We are upset we were put in this position."
Washington's contract was extended last year for 2010 before the drug test.
He met with his players earlier yesterday and told them about testing positive in July.
"He was very emotional. You could tell that he's a broken man from this one bad choice he made," Josh Hamilton said.
Hamilton has a long history of drug abuse and was suspended for the 2004 season while in the Rays' minors. He said there were no parallels between his crack cocaine habit and Washington's admission of one-time use.
"I was addicted to drugs,'' Hamilton said. "This was something of a weak moment, a decision of choice . . . Our stories are nothing alike.''
Washington, who has been subject to increased testing, said he passed every subsequent test. He said he has completed the MLB drug treatment program.
Management has a different set of drug-testing rules than those for players. Treatment is madatory for management employees who test positive for cocaine and other recreational drugs. Decisions on discipline are made by the team and MLB on a case-by-case basis.