Kinsler's second leadoff homer in three games was the eighth of his career, putting Texas ahead to stay in a 5-1 victory over the Chicago White Sox on Sunday night and tying the team record already shared by Michael Young and two others.
"Cool," Kinsler responded when told of the record.
"I didn't even know I had that record," Young said. "It's always big when you have your leadoff guy setting the tone."
"It's a really good feeling to be leading 1-0 after one at-bat," said Kinsler, who has three of his nine homers this season leading off. "It's just something that kind of happens. It's not like I'm going out there trying to hit a home run my first at bat."
Kinsler, an All-Star second baseman last season, is early in his fourth season and has led off 173 of his 395 career games. The other Rangers with eight leadoff homers are Mike Hargrove and Oddibe McDowell.
Danks, Texas' first-round draft pick in 2003, went 5 1/3 innings and gave up 10 hits for the first time in his 64 career starts -- all for the White Sox since being traded. It was the second straight game in which he allowed five runs.
Danks is 1-2 in three career starts against Texas, all of those coming at Rangers Ballpark for the Texas native who was the ninth overall pick.
"It's always been just another start. I've got more at stake than wondering what these guys are doing or thinking or trying to impress someone," Danks said. "I have my own career to worry about and I don't care what these guys think."
Danks was traded to the White Sox in December 2006 for Brandon McCarthy, who beat Chicago on Saturday night.
Harrison escaped a bases-loaded jam with the Rangers up 3-0 in the fifth by retiring Carlos Quentin on a flyout with his 100th pitch. Chicago had runners on in every inning against him.
"He made pitches when he had to make pitches," manager Ron Washington said. "They made him work out there today. ... But it was five pretty good innings because he kept them off the board."
Quentin's flyball capped a nine-pitch at-bat that first baseman Chris Davis tried to end a few pitches earlier when he fell into the front row of the new luxury seats chasing a foul ball. Davis jumped back onto the field unhurt.
"We didn't get anything done early with the bat," White Six manager Ozzie Guillen said. "That was the game right there, bases loaded and couldn't do anything."
Elvis Andrus, the Rangers' 20-year-old rookie shortstop, already had two hits and a stolen base before his sacrifice bunt in the fifth pushed Kinsler into scoring position. Andrus was hitting second after being in the No. 9 spot his first 18 starts.
White Sox OF Jermaine Dye was out of the lineup after being hit on the top of the left hand with a pitch Saturday night. X-rays were negative and Guillen said Dye could play Monday at Kansas City. ... Chicago 3B Josh Fields was hit on his left hand in the first inning, but stayed in the game. Guillen said he was fine. ... Hall of Fame receiver Michael Irvin, a three-time Super Bowl champion with the Dallas Cowboys, threw out the first pitch. It was wide, bouncing in the batter's box.