The Rays announced Tuesday that the surgery was performed by Dr. James Andrews on the 24-year-old Moore, who made the AL All-Star team last year.
Moore left his start on April 7 in Kansas City due to elbow soreness, and was placed on the 15-day disabled list the following day.
Also Tuesday, Tampa Bay activated reliever Juan Carlos Oviedo from the 15-day disabled list. Oviedo had right elbow ligament replacement surgery in September 2012.
Oviedo last pitched in the majors on September 21, 2011, for Miami, when he was known as Leo Nunez.