Chicago took three of the four games in the series.
"Just the way it occurred," manager Mike Quade said. "We haven't played great baseball, but there were a lot of plays today. ... There was some execution today that I really liked to see."
Fukudome singled and scored in the first inning, hit an RBI triple and scored in the second and delivered the big blow in the sixth when he connected off reliever Daniel Herrera to make it a four-run game.
The right fielder also threw out Rickie Weeks twice trying to stretch singles into doubles, so it wasn't hard to see why Quade called it a career day for him.
Fukudome shrugged that off, but he didn't downplay taking three of four from the NL Central leaders.
Considering the Cubs had dropped 11 of 13 before this series and began the day 13 games under .500, that was hard to envision.
"It definitely gives the good rhythm of winning," he said.
Even so, the Cubs got plenty of help in this one, with Weeks' ill-advised decisions on the bases in the fourth and seventh along with shaky defense and a poor performance by Greinke.
"I talked to Rickie," manager Ron Roenicke said. "I like these guys to be aggressive, but I just told him when we get down on runs and we're late in the ballgame, that's when we need to make sure of what we're doing. He's played awesome. He's on base all the time. He's played awesome."
Greinke was far from awesome.
The 2009 AL Cy Young Award winner allowed a season-high eight runs and eight hits in 5 1/3 innings and took his first loss since his Milwaukee debut against the Atlanta Braves on May 4, even though he struck out 10 and walked two.
The Cubs scored two runs against Greinke in the first and three more in the second, with three of those five runs coming on a passed ball, a wild pitch and an error, and things didn't get any better for Greinke after that.
He left trailing 6-5 with runners on second and third in the sixth after striking out Reed Johnson.
Herrera came in and Fukudome quickly gave the Cubs some breathing room when he drove a 1-2 pitch to the right-field seats for his third home run. That made it 9-5 and helped Garza (3-6) pick up a win even though he was far from dominant.
After missing three starts because of a bone bruise on his right elbow and then dropping two straight, he tied a season high by allowing five runs in six innings.
"That's a good team they have over there," Braun said. "I don't think they've played up to their expectations to this point. They're going to be in it at the end of the year, they've got a real good team."
Now, all the Cubs have to do is prove it. Soriano, who came off the disabled list Wednesday after being sidelined by a strained left quadriceps, notices a change.
"People are more relaxed, more enjoying it," he said. "I think that fuels us."
Quade wasn't exactly shedding tears over New York's Derek Jeter missing this weekend's series. The Yankees are making their second regular-season trip to Wrigley Field, but they won't have their captain. Six hits shy of 3,000, Jeter is sidelined by a strained right calf. "I'm like every other baseball guy. I don't want this guy coming here getting his 3,000th hit," Quade said. ... Roenicke said the Brewers sent left-hander Zach Braddock to Triple-A after Wednesday's game because of some punctuality issues. He was showing up at the ballpark on time but was late going out for pregame warm-ups. "Zach's a good guy and works hard, he's trying to manage his stuff," Roenicke said. "You need him to go somewhere he can get everything right. Once he gets everything right ... he's throwing the ball good so we'd like to see him back." Braddock has spent time on the DL this season because of a sleep disorder.