CLEVELAND -- Scott Kazmir can be excused for not recalling any details about his last major league victory.
"I can't remember that long ago," he said.
A two-time AL All-Star who pitched in an independent league last season, Kazmir (1-1) allowed two runs in six innings. It was his first win since beating Tampa Bay on Sept. 19, 2010, while with the Angels.
The 29-year-old Kazmir overcame a series of arm injuries and a stint with the Sugar Land Skeeters of the Atlantic League in 2012.
"I don't need to be reminded of it anymore," Kazmir said. "I'm glad this is out of the way so I don't have to worry about it anymore."
Cleveland has won six straight for the first time since winning seven in a row April 26-May 3, 2011.
Kevin Correia (3-2) failed to go seven innings for the first time in six starts this season. Rookie outfielder Aaron Hicks, the Twins' top prospect, hit his first major league home run to lead off the fifth, clearing the 19-foot wall in left field.
Kazmir, who allowed eight earned runs in 8 1/3 innings in his first two starts, struck out seven.
"This dude was pitching independent ball last year," Swisher said. "For him to be where he is now, he's got a lot left in the tank. I'm glad he's on our side."
Kazmir won 53 games for the Rays from 2005-09 before his career was derailed. He was 9-15 for the Angels in 2010 and appeared in only one game in 2011 before being released.
Kazmir was 3-6 with a 5.34 ERA in 14 games for Sugar Land last season and signed a minor league contract with the Indians in January. He won a spot in the rotation in spring training, but ended up on the 15-day disabled list before the season started with a pulled muscle in his rib cage.
"He's an easy guy to pull for," Indians manager Terry Francona said. "He loves to pitch. He wants so bad to make this work. We're pulling for everybody. That's the way we are, but it's hard not to have a soft spot when you see what he's been through."
Kipnis, who has six RBIs in his last two games, and Swisher both homered to right in the first. Swisher, who entered the game in a 4-for-28 slump, missed the two-game series against Philadelphia and didn't play Friday.
"You don't want to show up in the lineup and not pull your weight," Swisher said. "These guys have been on fire. As soon as you get back in the lineup you have to do your part as well."
Correia, who had won three consecutive starts, allowed four runs in five innings.
The Twins scored in the second on Pedro Florimon's RBI single, but his misplay in the field helped the Indians add two in the bottom of the inning. After Michael Brantley's single gave Cleveland a 3-1 lead, Florimon was unable to catch Kipnis' popup that fell for a run-scoring single. The shortstop ranged into shallow left field, but the ball drifted away from him and dropped in.
Twins manager Ron Gardenhire knows the Indians are finding ways to win.
"It's about finding the outfield grass and getting the balls through," he said. "It's not about how you hit 'em and how hard you hit 'em. And they did that. They popped some balls, they rolled them through. They ended up getting a couple more runs and kind of hung on from there."
Hicks, who is batting .123, homered on Kazmir's first pitch of the fifth. Joe Mauer had a hit and a walk in three at-bats and is in a 5-for-42 slump.
Indians setup man Vinnie Pestano was unavailable because of a "cranky" right elbow, according to Francona. Pestano threw a bullpen session before the game, but said he could go on the disabled list if the elbow doesn't feel better in the next day or two. ... Hicks leads AL rookies in walks with 13. ... Twins OF Oswaldo Arcia didn't start Saturday after starting 12 consecutive games. He had a pinch-hit single in the seventh. ... Mauer was the DH after catching 10 innings Friday. ... Indians OF Michael Bourn (lacerated right index finger) took batting practice Friday for the first time since being injured on April 14. Bourn could begin a minor league rehabilitation assignment in the next few days. ... Francona said RHP Brett Myers (sore elbow) threw off flat ground Saturday and is making progress to throwing off the mound . ... The Indians' April 23 rainout in Chicago has been rescheduled as part of a traditional doubleheader on June 28. .... The series ends Sunday when RHP Mike Pelfrey (2-3) faces RHP Corey Kluber (2-0).