Kluber (2-1), the 2014 Cy Young Award winner, entered the game with a 6.38 ERA, but was in command as he dealt the White Sox their second straight lost and third in their last four games. The right-hander struck out nine and walked two in his fourth career shutout and 11th complete game. It was his sharpest performance of four starts this season.
Only one Chicago baserunner reached second base.
Chicago's Jose Quintana (0-4) allowed three runs on five hits in six innings while striking out six and walking three. A first-time All-Star last season, the left-hander also entered the game with a bloated ERA, at 6.75.
Quintana was solid in this one, but lost his fourth straight start as his teammates couldn't solve Kluber. Three White Sox relievers combined for three scoreless innings and allowed just one hit.
Cleveland's Jason Kipnis played his first game this season, getting the start at second base and batting sixth after being activated from the 10-day disabled list. He went 0-for-3 with a walk.
The 30-year-old two-time All-Star was limited to two exhibition games with a right shoulder strain, then played seven rehab games in the minors.
White Sox third baseman Todd Frazier returned after sitting out four games with the flu and was hitless.
Cleveland jumped ahead 1-0 in the third when Austin Jackson slid home from third on a wild pitch. Jackson led off the inning by lining a double to left-center.
Guyer's homer, with Kipnis on first following a walk, made it 3-0 in the fourth with two outs. He drove Quintana's fastball into the left-center stands.
Kluber got some help from shaky White Sox base running to help preserve the lead. Melky Cabrera led off the seventh with a drive to the right-field wall, but was thrown out at second base after hesitating rounding first.
Moments before Friday night's first pitch, the White Sox placed right-handed starter James Shields on the 10-day disabled list, retroactive to April 18, with a strained right lat. Shields (1-0. 1.62 ERA) had been scheduled to start Saturday's game against the Indians, but felt discomfort in his last start against Minnesota on Sunday and again on Friday when he threw.
The 35-year-old righty underwent an MRI on Friday and will probably miss two or three turns in the rotation, according to GM Rick Hahn.
The White Sox purchased the contract of right-hander Mike Pelfrey from Triple-A Charlotte on Friday night and will insert him in Shields' rotation slot on Saturday.
NOTES: Indians manager Terry Francona said 2B Jason Kipnis can play without limitations. To make room for him, the Indians assigned INF Yandy Diaz to Triple-A Columbus. Diaz batted .236 with nine runs scored in the first 15 games of the season with Cleveland. ... RHP Mike Pelfrey, 33, was 0-2 with a 7.50 ERA at Charlotte. In 10 seasons in the majors, the 6-foot-7 right-hander is 65-91 with 4.57 ERA. ... Chicago C Geovany Soto (right elbow), who went on the 10-day DL retroactive to April 12, is making progress with his throwing, manager Rick Renteria said. ... White Sox LHP Carlos Rodon (upper bicep bursitis) will be in Chicago for an evaluation this weekend, but no timetable has been set for his return to the rotation.