The Tigers have won 12 straight games and lowered their magic number to clinch their first division title since 1987 to two.
"It shows how well we're playing right now," Raburn said about the streak. "Everything is coming together, pitching and offense. That's the key to winning ballgames."
Sergio Santos (4-5) was charged with the tying run in the ninth and then gave up the deciding run in the 10th to take the loss.
Alexei Ramirez had two hits, two RBI and a run scored in Chicago's fourth consecutive defeat.
Brad Penny tossed six full frames in the start for Detroit, allowing five runs -- one earned -- on eight hits and three walks, and Dylan Axelrod, making his first career start, allowed two runs on six hits and two walks while striking out eight through six innings for Chicago.
"He threw great. He deserved to win," White Sox catcher A.J. Pierzynski said about Axelrod. "For his first big league start to go out and pitch the way he did, it was fun to watch and nice to see."
Trailing by three runs entering the ninth, Raburn hammered a pinch-hit, solo home run off of Chris Sale. Sale then walked pinch-hitter Magglio Ordonez to bring the tying run to the plate with one out.
Santos took the bump next, and pinch-hitter Avila sent Santos' second pitch over the center field wall to tie the game.
After Pierzynski grounded into an inning-ending double play in the bottom of the ninth with a runner on third, Detroit scored the game-deciding run in the top of the 10th.
Victor Martinez slapped a one-out double just fair down the third-base line before being replaced by pinch-runner Will Rhymes. Guillen came up next and hit a Santos fastball back up the middle, easily pushing Rhymes across the plate.
Detroit struck first on Guillen's leadoff home run in the second inning.
Alex Rios led off the home half of the frame with a base hit and Alejandro De Aza followed with a walk. Brent Morel then laid down a sacrifice bunt, but Penny dropped the ball as he tried to bare hand it and Morel reached safely on the error, loading the bases.
Penny almost got out of the jam by striking out Gordon Beckham and getting a weak grounder to first by Juan Pierre that Miguel Cabrera threw home for the force. But Ramirez blooped a two-run double to shallow right before Paul Konerko lined a single to center that scored Pierre and Ramirez to put Chicago ahead, 4-1.
The teams traded runs in the sixth on Martinez's RBI single and a run-scoring double by Beckham.
Detroit's win streak is the longest by the club since the 1934 team won 14 straight...Konerko's two-run single in the second gave him his 99th and 100th RBI of the year, marking his sixth 100-RBI season of his career...The Tigers won the season series between the teams, 13-5...Chicago is 8-for-59 with runners in scoring position and have left 53 men on base over its last six games.