Street, who has converted all 21 saves opportunities this season, was put on the DL before Saturday night's game against the Pittsburgh Pirates. Right-handed reliever Cory Burns was recalled from Triple-A Tucson.
Street was injured in the ninth inning of Friday night's 9-8 win over the Padres when he went to cover first base on a game-ending groundout.
"I was hoping I wouldn't go on the DL but I'm going to miss at least two weeks and I'm hopeful that will be it," Street said.
Street is 2-1 with a 0.75 ERA in 36 games. He has not allowed a run in his last 22 appearances and 21 1/3 innings.
Street was the Padres' lone representative in the All-Star Game last month.
San Diego manager Bud Black said he likely would use a combination of set-up relievers Dale Thayer and Luke Gregerson to close games. Thayer was 5 for 5 in save opportunities in May while Street was on the DL with a strained right shoulder.
"It's a bummer," Black said of Street being injured. "I really believe the guys in our bullpen will step up, though. We have five rookies down there but we'll find a way to get it done until Huston comes back."
Burns was 1-2 with three saves and a 3.14 ERA in 54 games with Tucson. He pitched in one game with San Diego earlier this season, working 1 2/3 scoreless innings on Aug. 4 against the New York Mets in his major league debut.
Information from The Associated Press was used in this report.