Greene, who started at third base, had hit a solo home run in the second inning before he was plunked by Meche in the top of the fourth.
Greene said he didn't think that Meche was intentionally throwing at him.
Cardinals manager Tony La Russa said he was optimistic that Greene would be available on Monday when the Cardinals travel to New York to play the Mets.
"He got hit right in the point of the knee, and it's a very vulnerable spot," La Russa said. "So we'll see how sore he is tomorrow."
Greene's exit capped an eventful weekend in Kansas City. He returned from the disabled list on Thursday and made his first career start at third base on Friday against the Royals.
He had a home run and double in that one and added another homer on Saturday, before his third home run of the series on Sunday.
Greene hadn't homered in three consecutive games since he did it for the Padres from Sept. 9-11, 2004, against Colorado.
Greene left Friday's game early after fouling a ball off his left foot.
Rustin Dodd is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.