CLEVELAND -- The Chicago White Sox finally found the formula to win on the road.
Not only did the White Sox break their five-game losing streak, they won for the first time this season away from U.S. Cellular Field.
Chicago began the day as the majors' only winless team on the road. The White Sox were swept three games in Washington before dropping the first two against the Indians. Chicago avoided its first 0-6 start on the road since 1968.
"In this five-game skid, we've had a chance to get that big hit and make that big pitch and we just haven't been able to do it," Peavy said. "It's not like we've been getting boat-raced every night."
Peavy (2-1) gave up a home run to Michael Bourn on his first pitch of the game -- a line drive that landed in Chicago's bullpen -- but held the Indians to five hits overall and didn't walk a batter.
"I didn't expect Bourn to swing there," Peavy said. "He got me."
Indians manager Terry Francona gave full credit to Peavy, who retired 12 straight batters at one point and improved to 4-0 in five career starts against Cleveland.
"Bournie kind of ambushed him the first pitch of the game, but after that we didn't have many opportunities to score until late," Francona said. "He's just a really good veteran pitcher."
A day after watching left-hander Chris Sale allow a career-high eight runs in 4 1/3 innings, White Sox manager Robin Ventura liked what he saw from Peavy.
"He was just sharp today," Ventura said. "It was one of those (games) where he just seemed to be hitting the glove."
Bourn left in the eighth with a cut on hand that needed five stitches to close and is expected to sideline him for several games.
First baseman Adam Dunn fielded Bourn's slow grounder and tossed to pitcher Matt Thornton, who was covering on the play. Bourn made a headfirst into the bag for an infield hit, but Thornton stepped on his hand.
Francona and a trainer examined Bourn, who was removed because of the cut on the index finger of his right hand.
Konerko's two-run homer ruined a solid effort from Brett Myers (0-2), who took a shutout into the sixth. Cleveland's right-hander allowed 14 earned runs, including seven homers, in 10 1/3 innings in his first two appearances, but held the White Sox to four hits through five.
The home run was the second of the season for Konerko, who has hit 30 or more home runs seven times in his career.
"I feel like I'm climbing a little bit," he said. "I feel better. I wouldn't say I'm clicking on all cylinders, but just trying to grind through here. Early on, you're playing in tough weather, tough pitching and all that stuff. And it doesn't come easy."
Said Peavy: "We knew it was a matter of time that somebody was going to come up with a big hit. Who better than our captain?"
Konerko has 16 career home runs and 75 RBIs at Progressive Field.
The White Sox have made a habit of playing close games in the early going. Chicago has played 12 games, with 11 being decided by three runs or less. The exception was Saturday's 9-4 loss.
"It looked like after that first few innings it was going to shake out to be a tight ballgame, as it was," Peavy said. "We were fortunate enough to come out on the right side of things."
Giambi made his first appearance for the Indians, starting as the designated hitter and going hitless in four at-bats. He began the season on the disabled list with a strained back and was activated Friday.
The Indians again played without second baseman Jason Kipnis (sore left elbow) and catcher Carlos Santana (bruised left thumb) in the starting lineup. Kipnis was also out on Saturday while Santana, who walked as a pinch-hitter in the ninth, had missed three straight games.
The Indians are off Monday and begin a three-game series Tuesday against Boston at Progressive Field. Francona will face the Red Sox, the team he managed for eight seasons, for the first time since being replaced following the 2011 season when Boston collapsed down the stretch. Francona led the Red Sox to World Series titles in 2004 and 2007.
Chicago, which is on a 10-game road trip, begins a four-game series in Toronto on Monday. The White Sox will face former teammate Mark Buehrle in the opener. RHP Gavin Floyd (0-2, 5.56) will start for Chicago. ... The White Sox placed INF Angel Sanchez (strained lower back) on the 15-day disabled list and purchased the contract of INF Tyler Greene from Triple-A Charlotte. LHP Charlie Leesman was designated for assignment to make room for Greene on the 40-man roster. ... RHP Ubaldo Jimenez (0-1, 6.97) will pitch the opener of the Boston series on Tuesday.