Martinez, 20, was 7-5 with a 3.47 ERA in 19 starts at Class A Great Lakes in the Midwest League. He's struck out 91 over 106 1/3 innings and walked 19.
The Cubs dealt Barney, a 2012 Gold Glove-winning second baseman, on Monday, and he was expected to join the Dodgers on Wednesday. Barney was batting .230 in 72 games with Chicago and had seen his playing time diminish with the arrival of rookie Arismendy Alcantara.
Carrie Muskat is a reporter for MLB.com. She writes a blog, Muskat Ramblings, and you can follow her on Twitter @CarrieMuskat. This story was not subject to the approval of Major League Baseball or its clubs.