"Obviously, this is huge," Tigers manager Jim Leyland said. "I'm not saying we can't lose another game, but it is too late in the season to make up many of them."
Detroit, which closed within three games of the wild card-leading New York Yankees, trailed 2-1 entering the ninth. Sheffield, who hadn't played since Aug. 21 because of an injured shoulder, went 0-for-3 with two strikeouts and a walk.
"I felt good, but my timing is still off," Sheffield said. "I haven't seen a curveball in a month."
Pinch-hitter Timo Perez reached on an infield single off Jenks leading off the ninth and advanced when the throw from second baseman Danny Richar went into Chicago's dugout for an error. Mike Rabelo sacrificed and Casey, another pinch hitter, dribbled a single through the drawn-in infield to score pinch-runner Cameron Maybin.
"I've known Timo a long time, and he's been in some big situations, so I know he's not going to make a mistake on the bunt attempt," Leyland said. "Once he got to third, then you go strictly on speed. Timo's fast, but it's apples and oranges to compare anyone to that kid."
Brandon Inge grounded to the catcher, Jenks (3-5) hit Curtis Granderson with a pitch and Polanco looped a single to left, triggering a celebration at the plate as pinch-runner Omar Infante came across.
"I made some good pitches out there -- maybe they were too good," Jenks said. "It's one of those frustrating days when you know you are making good enough pitches to get the outs, and they keep getting hits."
The Tigers closed to within six games of AL Central-leading Cleveland, which lost 10-3 at Los Angeles. Detroit has a three-game series with another wild-card contender, Seattle, this weekend.
"We're still in this thing," Casey said. "We're three back with a lot of games left, and we've got Seattle coming in. Plus we've got Sheff back, and that helps big time."
"We had the bases loaded and almost didn't score, but Paulie came through in the clutch," Chicago manager Ozzie Guillen said. "We felt good going into the ninth."
Bobby Seay (3-0) pitched a perfect ninth as Detroit got its second successive victory in its final at-bat against the White Sox following five consecutive losses to Chicago.
"There wasn't a feeling of depression after the eighth -- maybe that was because of last night," Inge said. "This was more like last year, where we knew we could pick Fernando up."
Inge drove in the game's first run with a third-inning home run. It was his 13th of the season but first since July 21.
"I didn't do anything different today -- I always think I can hit any pitcher," said Inge, who hit .209 in his prior 59 games before Thursday. "But that's still a good feeling when that ball gets out."
Mark Buehrle allowed one run and seven hits in seven innings for Chicago, which lost for the sixth time in eight games overall and the 13th time in 15 road games.
"I'm just going out and doing my job," said Buehrle, 0-3 in his past six starts. "I do whatever I can, and then turn it over to the bullpen. I just want one more win. That's all I'm begging for."
Durbin, making his first start since Aug. 18, gave up four hits in five scoreless innings.
- Chicago leads the season series 9-6. The teams will play three games in Chicago from Sept. 28-30 to close the regular season.
- Rodney has a 10.13 ERA in his past six outings.
- Jason Grilli pitched two scoreless innings for the Tigers, extending his shutout streak to 14.