CHICAGO (AP) -- This time, Chris Sale had to rely on Chicago's lineup to get the win.
Sale (3-0) pitched five innings in his first win against Cleveland since May 1, 2012. The All-Star left-hander allowed three runs and six hits while throwing 105 pitches.
Sale went 11-14 last year despite a 3.07 ERA. He also had four complete games, tying Tampa Bay star David Price for the AL lead.
But he got a boost from his offense in his third start of this season. Leadoff hitter Adam Eaton had two hits and scored three times as the White Sox picked up their ace on a sluggish night.
Chicago has scored 37 runs in its five home games this year.
''I think if we keep doing what we've been doing and making strides in the right direction we can be a force, be a great team,'' Sale said. ''It's fun to watch, I got to watch a lot of it tonight unfortunately.''
The White Sox have captured the first two games of their first series of the year against the Indians, who had a 17-2 record against Chicago last season. They play again on Saturday and Sunday.
Indians starter Carlos Carrasco (0-2) was tagged for five runs and six hits in 4 2-3 innings. It was another shaky start for the right-hander, who allowed five runs in 5 2-3 innings of a 7-3 loss to Minnesota in his season debut.
Cleveland pitching issued nine walks, including walking in two runs.
Gillaspie hit a sacrifice fly in Chicago's two-run third inning, helping the White Sox build a 3-0 lead. Adam Dunn, who had two hits and walked twice, singled and scored on Alejandro De Aza's base hit in the fourth.
Sale, who went 0-4 with an 8.61 ERA in four starts against Cleveland in 2013, was unable to protect the lead. Asdrubal Cabrera had a two-run double in the fifth, and then scored the tying run on Nick Swisher's single.
But Cleveland's pitching struggled throughout the game.
''When you score and tie it up, you gotta go back out and take charge of the game,'' Indians manager Terry Francona said. ''They just kept scoring every inning.''
The White Sox averaged 3.19 runs a game in Sale's starts last season.
''These guys are always tough on him,'' White Sox manager Robin Ventura said. ''They were tough on him last (year) and for whatever reason they can foul off a lot of pitches. They probably see something off of him. I don't know what it is. You don't see that many teams foul that much off. He got up there with a high pitch count.''
The White Sox grabbed the lead for good in the fifth.
Gillaspie singled in Eaton, and Carrasco was lifted for Marc Rzepczynski after striking out Jose Abreu for the second out of the inning. Rzepczynski walked Dunn to load the bases and then walked in a run as Dayan Viciedo came back from an 0-2 count to give the White Sox a 5-3 lead.
''I thought Carlos in the fifth kind of slowed down his tempo, you could tell he was fighting some of his mechanics,'' Francona said. ''That's when we went to Zep and tried to snuff that out. Nothing, we couldn't put up a zero the rest of the way.''
Gillaspie committed an error in the sixth, allowing Michael Brantley to score, but he responded with a two-out, two-run double in the bottom half, making it 7-4. Alexei Ramirez added an RBI single for Chicago in the seventh.
NOTES: OF Michael Bourn went 1 for 5 with a stolen base in his second rehab start for Double-A Akron. Francona said Bourn, who has yet to play this season with a left hamstring strain, would be examined after the game and though Bourn would prefer to return to the big leagues Sunday, Francona said the team may hold off due to the wet conditions expected that day. ... Indians DH Jason Giambi (broken rib) made his first rehab start for Akron. He was 0 for 2 with two strikeouts. ... The White Sox acquired OF Gorkys Hernandez from Kansas City for cash considerations and assigned him to Triple-A Charlotte.