Nevin also figures to be a backup third baseman and pick up some playing time in the outfield.
He said he wants to see Lee back in the lineup as soon as possible.
"I'm a Cub now, and I want to win, and for this team to win, Derrek needs to be back," Nevin said. "I hope he comes back next week and makes us better. I'll do what they ask me to do."
Nevin was used as a designated hitter for Texas before Wednesday's trade, which sent Jerry Hairston Jr. to the Rangers.
"It's going to be nice to wear a glove again," Nevin said. "I missed that. The role I was in over there -- I didn't get to play any defense. It's kind of a different gig."
REPLAY: After suffering through a 7-22 month of May, the Cubs were hoping for better days ahead in June. It seemed like a month for their first win in June as they beat St. Louis, 5-4, in a five-hour, four-minute battle.
Todd Walker's groundout in the 14th brought home Juan Pierre for what turned out to be the winning run. Ryan Dempster, the eighth Cubs pitcher used on the night, pitched a scoreless bottom of the inning and the Cubs improved to 21-32.
The Cubs trailed 4-2 heading into the ninth, but Cardinals' third baseman Scott Rolen made an error on a Walker grounder with the bases loaded to knot the game at 4-4.