The Angels announced the move Thursday.
Street injured himself while warming up Wednesday. The 32-year-old veteran has converted all five of his save opportunities this season with a 1.17 ERA, allowing just four hits and one run over nine appearances.
"I made one more throw and thought, 'There's definitely something in there,'" Street said, according to The Orange County Register.
The Angels are uncertain how long Street will be out. Their new closer will be eighth-inning specialist Joe Smith, who earned the save Wednesday night when Los Angeles completed a three-game sweep of defending World Series champion Kansas City.
The Angels will call up a reliever on Friday. They have the day off Thursday before opening a six-game road trip at Texas.
Information from The Associated Press was used in this report.