Cleveland took two of three from scuffling Chicago and has won 11 of 16 overall. It begins a three-game series at the AL Central-leading Royals on Friday night.
"The thing is, try not to look too far ahead," Bourn said. "That's the biggest key -- don't focus on what everybody else is doing every day. All we can control is what's in (the clubhouse), then go play."
The Indians trail the Royals by 5 1/2 games, with Detroit in second place. But the Indians have six games left against Kansas City and seven against Detroit.
"We're going to fly the same flight," manager Terry Francona said. "We're going to eat the same food. I mean, it's exciting. Shoot, (the Royals) are playing great, so we don't need to get ahead of ourselves."
Carrasco (6-4) improved to 3-0 with a 0.73 ERA in four starts since he returned to the rotation. The right-hander also began the year as a starter, but then lost three of his first four outings.
He struck out seven in 6 2/3 innings while holding the White Sox to one run and four hits.
"You know, that's what I've been doing the last few games -- attack the hitters," Carrasco said. "I just need to do my job and keep the team in there."
Cleveland grabbed the lead for good in the sixth. Jose Ramirez led off with a single and stole second. Then he advanced to third on a groundout and scored on Kipnis' base hit, giving the Indians a 2-1 advantage.
Brantley added an RBI single in the seventh.
Danks allowed two runs and five hits while falling to 0-3 with a 6.75 ERA in five August starts.
"He's pretty good," Francona said when asked about the closer's dominant performance. "You may need to get cable or something to watch him."
Cleveland jumped in front when Bourn tripled on the first pitch of the game and scored on Jose Ramirez's grounder.
LIKE OLD TIMES
Francona said he keeps track of the Chicago Cubs when possible. Francona and Cubs executive Theo Epstein were part of two World Series titles with the Red Sox.
"I know they have a lot of young kids coming and they're pretty excited about it," said Francona, who has exchanged voice mails and text messages with Epstein in recent months. "I'm not surprised. To anybody who ever asked me about Theo when he came to Chicago, the two things I've said were, he won't get outworked and he's too smart not to figure it out."
SAFE AT HOME
Kipnis hit safely for the 25th time in 27 career games at U.S. Cellular Field. The Chicago-area native also escaped serious injury when White Sox third baseman Conor Gillaspie's heel inadvertently clipped him in the face.
Indians: C Yan Gomes is on pace to return from the seven-day concussion list before the series opener in Kansas City on Friday. As a precautionary measure, Gomes will take part in the pregame workout before the roster move is completed. ... 1B Nick Swisher is expected to spend most if not all of September in Los Angeles, where he is recuperating from recent surgeries on both knees.
White Sox: OF Moises Sierra continues to work his way back from a strained left oblique. He was 4 for 12 in the first three games of his rehab assignment with Triple-A Charlotte.