Texas also claimed right-hander Darren O'Day off waivers from the New York Mets, and the reliever will take Benson's spot on the roster.
Benson, who threw a bullpen session on Wednesday at Rogers Centre and said he felt well, will now miss his scheduled start against the Orioles on Saturday. Right-handed reliever Scott Feldman will start the game.
Benson is being sent back to Texas to visit team physician Dr. Keith Meister and will rejoin the club during its series with Baltimore. It is not certain when Benson will be able to pitch again, though.
Manager Ron Washington said on Wednesday that he doesn't think Benson's stint on the DL will last for an extended period of time.
"This is just elbow soreness," Washington said. "[Benson] actually said he could take his start. This is us being precautionary."
Benson -- who was signed by Texas as a Minor League free agent at the start of Spring Training -- said he originally felt the soreness in his elbow during a start at the end of the spring. He then tried pitching through it during his first two starts of the season, before deciding to skip a start this week in order to rest his elbow. He is 1-1 with a 9.00 ERA this year.
Washington said the 26-year-old O'Day will be used in middle relief out of the bullpen.
"He'll give us a different look out of the bullpen," Washington said of the sidearmer, who was recently designated for assignment by the Mets.
O'Day has also spent time with the Los Angeles Angels during his two-year career. He is 0-1 with a 4.27 ERA across 46 1/3 career innings.
To make room on the 40-man roster for O'Day, right-hander Dustin Nippert has been transferred to the 60-day DL from the 15-day DL. Nippert has been on the DL since the start of the season with a strained rhomboid muscle. He threw in the bullpen on Tuesday in Oklahoma City and experienced recurring tightness.
David Singh is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.