"According to our medical people, who were in contact with Dr. Andrews, it was a successful surgery," manager Bud Black said.
Kelly, 23, appeared in three games during Spring Training and felt discomfort in his elbow following his start on March 7. He then had extensive tests that showed 'micro tears' in his ulnar collateral ligament.
Kelly made his Major League debut last season for the Padres, going 2-3 with a 6.21 ERA in six starts. He came into camp last month as one of several candidates trying to win a rotation spot. He posted a 7.71 ERA over seven innings.
Kelly began last season with Triple-A Tucson, then missed three months with a right elbow strain. He eventually got healthy and reached the Major Leagues in August.
"In 2014, we'll put all this behind us and come back and be healthy again," Kelly said last month after it was determined he needed surgery.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.