Sunday was a step in that direction.
"We're gonna pick it up," said Anderson, who is the White Sox shortstop. "A win like this is definitely something we can build off of."
Francisco Lindor homered for Cleveland, which had its five-game winning streak end.
Holland (2-2) allowed one run on three hits in six innings. He walked three and struck out six in lowering his earned run average to 1.99.
"He was mixing up speeds, hitting his spots, coming hard in to righties," Indians left fielder Brandon Guyer said. "He had good stuff today, pretty similar to the last time we faced him."
Indians starter Danny Salazar (1-2) allowed four runs on seven hits in five innings.
Following back-to-back three-hit shutouts, the White Sox broke through immediately Sunday.
Anderson led off the bottom of the first with a double and advanced to third on a single by Tyler Saladino. Cabrera then drove in Anderson with a sacrifice fly to left field to make it 1-0, snapping a 23-inning scoreless streak for Chicago. Cleveland pitchers also entered with a 23-inning scoreless streak.
Jose Abreu followed with a line-drive single and the ball scooted past right fielder Abraham Almonte -- allowing Saladino to score from first base and Abreu to advance to third. Two batters later, Leury Garcia singled to drive in Abreu and make it 3-0.
"I didn't have that great a feeling with the change-up the whole game," Salazar said. "I try to do the best I can out there. Not thinking about (the error). Abraham has made some amazing plays for me out there."
Cleveland got a run back in the fourth on a lead-off home run by Lindor.
In the top of the fifth, Austin Jackson led off with a double and Almonte followed with a walk. Both runners then advanced a base on a wild pitch. But Holland struck out three of the next four batters -- working around a two-out walk to Carlos Santana to load the bases -- to escape the jam. A strikeout of Lindor ended the threat.
"It was a big jam because he had some guys coming up there that could do some damage," Chicago manager Rick Renteria said.
The Indians entered the game with a total of seven errors on the season, but had three on Sunday in a sloppy performance.
"It makes it hard when you're playing like that," manager Terry Francona said. "We didn't finish some plays. We kicked around the ball a little bit."
KING OF CLEVELAND
Holland, an Ohio native, is now 7-1 with a 2.27 ERA against the Indians in his career.
"I'm not going to think anything of that," he said. "No matter what, that's an unbelievable team over there. They've done some great things for the state of Ohio. I've got a lot of friends and family back there that's pulling from them, except for when they're playing me."
White Sox: RHP Zach Putnam is listed as day-to-day after leaving Saturday's game with soreness in the back of the elbow. "Putnam is available," Renteria said before the game. "But would I use him? Probably not." . LHP Carlos Rodon (bursitis in left biceps) and Jake Petricka (strained right lat) both returned to Chicago this weekend to be evaluated before the next phase of their rehab is determined.
Indians: Following a day off, they begin a three-game home series against the Houston Astros Tuesday night with RHP Josh Tomlin (1-2, 11.68 ERA) taking on LHP Dallas Keuchel (3-0, 0.96) in the opener.
Denard Span turns a 1-run deficit into a 1-run lead with a double to right field in the eighth inning.
Ian Kinsler homered and drew a bases-loaded walk, Mike Trout had three hits and a sacrifice fly, and the injury-plagued Los Angeles Angels beat the Oakland Athletics 8-4 on Friday night to snap a four-game skid.
Four Houston Astros starting pitchers have won at least eight games this season. They are Gerrit Cole, Charlie Morton, Justin Verlander and Lance McCullers.
Enrique Hernandez sparks the Dodgers' offense in the first inning with a long ball out to left-center field.
In Carlos Martinez's first eight starts of the year, the St. Louis Cardinals right-hander pitched at a Cy Young Award level. While his record was only 3-2, his ERA was a miniscule 1.62 and he gave up only five earned runs in a seven-start stretch.
Albert Pujols hits a dribbler to pitcher Chris Bassitt, who airmails the throw to first and allows a pair of runs to score.