Johnson had the procedure performed at New York-Presbyterian Hospital by hand specialist Dr. Melvin Rosenwasser.
The 31-year-old designated hitter has been out since May 7 with what was originally diagnosed as an inflamed tendon in his wrist, which has previously been operated on.
"It's frustrating," Johnson said on Monday. "We'll get it fixed and I'll be able to swing pain-free, and we'll go from there."
Yankees manager Joe Girardi said on Monday that he does expect Johnson to return to the team late in the summer but cautioned that the surgery "is not a quick fix."
"I believe we're going to get him back, but I don't believe that we're going to get him back in four weeks or three weeks like we anticipated," Girardi said.
"It's the middle of May, so you're talking about the middle of June or July 1 before he can even pick up a bat. Then you're talking about some substantial time after that, because it's going to take him time to get his strength. We're looking at a while."
A career .270 hitter, Johnson was signed to a one-year, $5.75 million deal in December. He was batting .167 with two home runs and eight RBIs in 24 games when he was placed on the disabled list for the 10th time in his career.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.