Street was hurt playing catch before Wednesday's game against the Kansas City Royals.
The veteran right-hander, who said he had never before injured an oblique, tweaked the muscle to the side of the abdomen near the end of his throwing routine on a light, short toss.
"I made one more throw and thought, 'There's definitely something in there,'" Street said, according to The Orange County Register.
Setup man Joe Smith had to close out the Angels' 4-2 victory.
"It's never a good thing when you lose your closer, but that's why there's seven of us out there [in the bullpen]," said Smith, according to The Orange County Register.
Street has five saves and a 1.17 ERA this season.
The Angels said a corresponding roster move will be made Friday, when the team returns to action against the Texas Rangers.