Rodriguez turned his ankle after rounding the first-base bag on a single in the seventh inning Wednesday night and had to be helped off the field in what became the Cubs' 6-4, rain-shortened victory.
X-rays taken at a local hospital showed no break, though Black said the sprain is severe enough that Rodriguez will be placed on the disabled list, probably Friday. The Padres will play with 24 men in Thursday's series finale, with Chris Burke likely to start at short.
The Padres have lost five straight and are 13-21 after starting the season 9-3. Their starting pitchers haven't won a game since April 16, and the offense is dead last in the Majors in runs scored. And now, the shortstop's hurt.
Black noted the presence of Burke, along with David Eckstein and Edgar Gonzalez as options "in a pinch," but indicated general manager Kevin Towers would be working on bringing in someone else.
"We're working our way through that right now," Black said with diplomacy. "K.T., I'm sure, as we speak, is doing everything he can to try to fill that void."
After lining a pitch to left against Chicago's Ted Lilly, Rodriguez slowed rounding first and appeared to bluff a move toward second. His left leg appeared to lock and he limped back to first base, just getting in ahead of Derrek Lee's tag.
Rodriguez was assisted off the field by first-base coach Rick Renteria, along with Padres strength coach Jim Malone and trainer Todd Hutcheson.
Rodriguez, who went 2-for-3, was replaced by pinch-runner Kevin Correia then by Burke in the field in the bottom of the seventh.
David Brown is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.