Danks (14-11), who gave up seven hits, fanned five batters and walked two to pick up his first win since September 4 for Chicago, which has won three in a row since an eight-game skid.
"I tried to finish on a high note," Danks said. "Guys made some good plays behind me and we scored six runs."
Chicago has won six in a row versus Los Angeles for the first time since April 7- June 10, 1992.
Scott Kazmir (9-15) lasted just three-plus innings in the start, allowing five runs on five hits. The left-hander is 1-6 over his last nine starts.
"He didn't have good command tonight and never got into a rhythm," Angels manager Mike Scioscia said of Kazmir. "We had a lot of opportunities with guys in scoring position."
The Angels scored their only runs of the game in the first inning. Peter Bourjos walked and raced home on Kendrick's double. Hunter hit an RBI single two batters later to make it a 2-0 game.
Chicago took the lead for good with a four-run third. Brent Morel singled and Brent Lillibridge walked. Both runners advanced one base on a sacrifice bunt. Alexei Ramirez followed with a two-run single, went to second on a passed ball and scored on Rios' double. Manny Ramirez capped the inning with an RBI double.
Castro's homer to center in the fourth made it a 5-2 game. The Angels stranded men on the corners in the home half.
Chicago added a run in the seventh for a 6-2 advantage. With a man on third, Pierre reached first and advanced to second on Rich Thompson's throwing error
The White Sox lead the season series with the Angels, 6-2...Chicago manager Ozzie Guillen ensured his fifth winning season with the club....Danks improved to 3-3 lifetime against LA...Kazmir fell to 3-5 in his career versus the White Sox.