Anderson scored the go-ahead run in the 10th inning on Raburn's third error of the game, lifting the Royals to a 3-2 victory over the Tigers on Sunday.
"We'll take anything right now," Kansas City manager Trey Hillman said. "Detroit made some mistakes, especially at third base, and we were able to capitalize."
The Tigers missed several chances to avoid extra innings, going 0-for-16 with runners in scoring position.
"I really have no problems, because we were trying to do the right thing," Detroit manager Jim Leyland said. "It's a shame, but you have to give them credit for not letting us get the big hit."
With one out in the 10th, Anderson reached on a dribbler down the third-base line. David DeJesus followed with a routine grounder to third, but the ball hit off Raburn's glove and went through his legs for his second error of the game.
Both runners moved up on Willie Bloomquist's comebacker to Fernando Rodney (2-3) and Billy Butler followed with another grounder. Raburn's throw to first pulled Miguel Cabrera off the bag as Anderson scampered home against his former team.
"Give Billy credit -- he's one of the slowest runners on our team, and if he doesn't hustle there, Cabrera might be able to tag him," Hillman said. "That's how you take advantage of mistakes."
Joakim Soria (3-0) threw two scoreless innings to get the win.
"I rolled the dice a bit -- that's the first time I've brought him into a tie game on the road," Hillman said. "I was committed to using him for two innings if we were winning, and with the middle of their order coming up, I figured it was time for him."
Detroit center fielder Curtis Granderson helped send the game to extra innings with a spectacular catch on Mike Jacobs' long drive in the eighth. Granderson raced back, leaped and twisted in mid-air to catch the ball as he slammed into the wall.
Jacobs singled in a run in the first against Armando Galarraga and Bloomquist added an RBI double in the fifth.
Magglio Ordonez homered in the second for Detroit, which finished with 10 hits. Adam Everett scored the tying run on Kyle Davies' wild pitch in the fifth, but the Tigers struggled offensively despite putting runners on base in all 10 innings.
"We don't have any excuses -- not at this part of the season," Cabrera said. "We want to win the division. Tomorrow is a good day off -- we can rest and think about what we did wrong today."
Davies allowed two runs and nine hits in six innings.
"They got a lot of hits, but it didn't seem like they were hitting the ball hard off me," he said. "I was efficient, and I was able to make pitches when I needed them. I just made the mistake to Ordonez and bounced the ball in the fifth."
Ordonez had Detroit's best chance to take control of the game, facing Jamey Wright with runners on second and third with one out in the eighth. Ordonez was ahead 3-1 in the count, but took back-to-back called strikes before Raburn flew out to end the inning.
"When it was 3-1, I knew I had first base available, so I was able to throw him a curveball," Wright said. "I threw him a good fastball on the outside corner, and he was definitely expecting something else."
Galarraga, making his first start in nine days due to illness, gave up two runs and four hits in five innings.
"He lost quite a bit of weight and he was weak," Leyland said. "He was too slow in the first inning. I told him to move it along a little bit or we'd have fans passing out from the heat and players passing out."
Detroit pitcher Dontrelle Willis (anxiety disorder) is scheduled to make a rehab start for Triple-A Toledo on Monday. Jeremy Bonderman (blood clot) and Nate Robertson (elbow) both made rehab appearances for the Mud Hens on Saturday. ... Sunday was the final game of Rick Porcello's five-game suspension for Tuesday's fight with Boston. He is scheduled to start for Detroit on Tuesday. ... The Royals have scored in the first inning in five of their last six games.