"I had a very weak moment," Washington said at a news conference on Wednesday. "I did wrong and I take responsibility for that and I'm sorry."
The 57-year-old manager met with his players earlier in the day and told them he tested positive in July.
Washington said he used cocaine only once and called it "stupid behavior."
Washington said he offered to resign after the drug test. He said he was grateful team president Nolan Ryan and general manager Jon Daniels kept him on.
Washington's contract was extended last year for 2010 before the drug test. His contract expires after this season, which will be his fourth with Texas.
SI.com said Washington informed the commissioner's office and team management about using cocaine before the results of his test came back. The Web site said Washington has been subject to increased testing since then, and has passed every subsequent test.