New York also had two players selected from its Triple-A roster by other teams during the Major League phase, with right-hander George Kontos taken by the Padres and right-hander Lance Pendleton going to the Astros.
Fish, 22, was taken from the Angels with the 28th pick and will have an opportunity to make the Yankees' roster as a reliever in the spring. He was 5-5 with a 6.79 ERA and two saves in 49 relief appearances with Class A Rancho Cucamonga and Double-A Arkansas this year.
Yankees general manager Brian Cashman said that Fish has a fastball, curveball and changeup that drew the organization's attention.
"We like his stuff," Cashman said. "We're going to see if it can translate for us at the Major League level."
Turpen, 24, was 7-6 with a 4.30 ERA and four saves in 49 relief appearances for Double-A Richmond in the Giants' system and Double-A Portland in the Red Sox's organization this year. Cashman said that Turpen throws his fastball and slider from a low three-quarters arm slot and induces ground balls.
"We're going to take a look at him, too, out of the bullpen," Cashman said.
During the Major League phase of the Rule 5 Draft, eligible players left unprotected from their clubs' 40-man rosters may be selected for $50,000. A player selected must remain on his drafting team's active Major League roster during the following season or be sent back to the original club for $25,000.
The Yankees did not gain or lose any players during the Minor League portions of the Rule 5 Draft.
Kontos, 25, was 0-4 with a 3.60 ERA in 24 appearances (two starts) at three levels of the system this year. Pendleton, 27, was 12-5 with a 3.61 ERA in 29 games (27 starts) at Double-A Trenton and Triple-A Scranton/Wilkes-Barre.
"I thought it was possible to lose Kontos and Pendleton," Cashman said. "I thought they were quality picks by the organizations that picked them, so we'll see what happens."
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.