Rogers has been on trial with the club over the winter break and even linked up with the squad for the 1-0 FA Cup third round defeat to Arsenal at the Emirates on Monday.
The 24-year-old wideman will now join the Championship side on permanent basis, with Leeds manager Simon Grayson revealing that he would be an immediate part of his first-team plans.
"We're delighted to have agreed a deal with Robbie," Grayson told the club's official website.
"He's been training with us and he came to Arsenal to get a feel of the group.
"He has been playing in the MLS where he has done well, he wants to continue his career in Europe, and we believe he will make an impact here.
"As soon the work permit business is complete he will be an immediate part of our plans and have an opportunity with the first team."Rogers has made over 100 appearances for MLS side Columbus Crew, and will be keen to help bolster Leeds promotion chances this season, with the cub currently sat eighth in the Championship table.
American players Mike Grella, Eddie Lewis and Eric Lichaj have all spent time at Elland Road.