Andrus, who turns 34 next week, recently cleared waivers after being designated for assignment by the Oakland Athletics. Andrus gives the White Sox experience with Anderson expected to be out until late September after having surgery for a torn ligament in his left middle finger on Aug. 11.
"I'm very happy to be here," Andrus said before the White Sox played at AL Central-leading Cleveland. "It's an amazing group of guys. We're in contention for the playoffs. That's why you play every single year, to have this opportunity."
The White Sox entered the weekend series trailing the Guardians by 2 1/2 games and second-place Minnesota by one.
"We all compliment the front office," Chicago manager Tony La Russa said. "Trying to fill Tim's shoes and you have somebody like Elvis, he's got a lot of game left. He plays the whole game. He still fields well. He can run the bases. He has a knack for getting hits against good pitchers. He played in a couple World Series. We're thrilled."
Andrus is a career .270 hitter with 324 steals in 14 seasons with the Texas Rangers and Athletics. He batted .237 with eight homers and 30 RBI in 106 games with Oakland and committed nine errors in 101 games at shortstop. The Athletics let Andrus go after opting for Nick Allen at shortstop for the rest of the season.
Andrus is hitting .342 (113-for-330) with 10 homers with 44 RBI in his career against Cleveland. He's even more productive at Progressive Field, hitting .378 (65-for-172) with seven homers and 24 RBI.
"I think every player has a team whenever you face them everything gets back to normal," Andrus said. "Even if you're struggling a little bit you see that team and your swing got better right away. For some reason I love hitting here. The ball looks so huge."
Chicago also optioned infielder Lenyn Sosa to Triple-A Charlotte. He is 4 for 35 with one home run in 11 games with the White Sox this season.