Manager Don Mattingly described the injury as "fairly major," "significant" and "not just a strain."
He said newly signed Bobby Abreu would get most of Rivera's playing time in left field, but Mattingly is "not looking at him as an everyday guy" because that would "run the risk of beating him up."
Rivera, who missed six starts last month with a left hamstring injury, was removed from Tuesday night's game when he felt "pins and needles" in the same area behind the back of his left knee while sliding into third base.
An MRI exam on Wednesday revealed the extent of the injury.
"I felt something pop, but it didn't hurt," said Rivera. "They said this is better than pulling a hamstring. It might take two or three weeks and they want me to be 100 percent."
Rivera said he first felt discomfort three weeks ago in Milwaukee and tried to play through it. He said the injury Tuesday night felt different but occurred in the same spot as the previous discomfort.
Van Slyke, 25, was hitting .336 with eight homers and 25 RBIs playing mostly in the outfield, but occasionally at first base. He is the son of former Major League outfielder Andy Van Slyke.
A 14th-round Draft pick in 2005, Van Slyke spent four years at Rookie-level and Class A before putting his game together, hitting at least 20 home runs in two of the past three seasons. He pinch-hit in the sixth, delivering a broken-bat RBI single.
"I wasn't expecting this, but it was cool," said Van Slyke, last year's Minor League position player of the year. "I wasn't as nervous as I thought. I got the first swing out of the way and after that, it was just another game."