The Brewers finally pulled out a close win after dropping five of the previous nine by one run, thanks to one dominant pitching performance and one clutch hit.
"I felt better as the game went on," Greinke said. "I was a little off and on early. Got some breaks. Some line drives hit at people and then I kind of settled in a little bit."
The Brewers ultimately rewarded him in the 10th.
Aramis Ramirez led off by lining his second double of the game down the left-field line against Jesse Crain (1-1). Nyjer Morgan ran for him and advanced on a wild pitch before Weeks drove a single past a diving third baseman Orlando Hudson to make it 1-0.
John Axford came on in the 10th and got his 13th save in 17 chances after blowing three of his previous five opportunities. He walked Gordon Beckham leading off, but struck out Adam Dunn before Paul Konerko grounded into a double play to end the game.
That gave Greinke (8-2) the win on a night when Sale pitched eight scoreless innings for the second time in three starts.
"You don't have long innings," Greinke said. "You just come in. You get your five, 10 minutes, just relax a little and you're back out there. Sometimes, you get in more of a rhythm when there are no long innings."
Greinke struck out four and walked one. The only hits he allowed were singles by Alex Rios in the fifth, Beckham in the seventh and A.J. Pierzynski in the eighth, and the White Sox hit into double plays in each of those innings.
Sale was about as dominant, striking out seven and walking one as Chicago lost for the seventh time in nine games.
"I see why his record (8-2) and his ERA (2.24) are where they are," Brewers manager Ron Roenicke said. "He's really funky along with great stuff."
White Sox manager Robin Ventura was impressed by both starters.
"You're looking at two pitchers that were ... just moving it in and out, back and forth," he said. "I think both of them were doing that all night. It's not just one pitch that they have."
Addison Reed pitched a scoreless ninth for Chicago thanks to two neat plays by Hudson at third. He made a leaping catch on Norichika Aoki's liner leading off and ranged far to his left on Carlos Gomez's grounder before Ryan Braun struck out.
Both teams were busy before the game.
The Brewers signed veteran right-hander Livan Hernandez to a one-year contract, after he was released by Atlanta.
"Obviously, having Phil go down, that's a big blow," Sale said. "We love having him in there on the mound and what he brings to our team is great."
The Brewers recorded their fourth shutout, while the White Sox failed to score a run for the seventh time. ... Braun extended his hitting streak to 16 games with a single to shortstop in the fourth. ... The Brewers also designated LHP Juan Perez and INF Edwin Maysonet for assignment and reinstated SS Cesar Izturis from the 15-day DL. ... C Jonathan Lucroy had a pin removed from his right hand on Thursday and could begin catching bullpen sessions in the next day or two. "He's raring to go," Roenicke said. ... With Humber out, Ventura said Dylan Axelrod will start Saturday against Randy Wolf. ... The White Sox purchased the contract of right-hander Brian Bruney from Triple-A Charlotte.