The 24-year-old USA international, who has played more than 100 games for Columbus Crew, has signed an 18-month deal with the Whites, with Leeds holding the option of a one-year extension to the contract.
Rogers was handed a work permit last week and, after returning to the United States to finalise his visa, he is expected to be training with his new team-mates over the weekend.
Having featured in a behind-closed-doors game against Newcastle last week, Rogers was forced to miss this week’s game against a Celtic XI.
United beat Celtic 2-0 at Thorp Arch, with goals from Matty Pattison and Billy Paynter.
South African trialist Pattison opened the scoring from the edge of the box. Paynter scored from close range late in the game.