Barney reached on a throwing error by shortstop Jed Lowrie to open the seventh, and two outs later, Lake smacked his second home run.
Chavez scattered seven hits over 5 2/3 innings in his longest outing this spring. He walked one and struck out six, and survived a ball off his leg by Ryan Kalish in the fifth.
Jason Hammel, making his third Cactus League start, was roughed up in the first. Jed Lowrie tripled with two outs in the Oakland first and scored one batter later on Josh Reddick's single to center. Alberto Callaspo followed with an RBI single to make it 2-0.
"I'm not too happy with it," Hammel said of the first inning. "Sometimes you're not going to have your best stuff and you have to find a way. We did that today. It definitely wasn't pretty."
Cubs pitching coach Chris Bosio told Hammel he was trying to do too much, was a little tentative and needed to keep the ball down.
"Strike one has to be the most important thing for me, and I'm learning that as guys know I'm a sinkerballer, they're seeing it down and leaving it down," Hammel said. "I'm making adjustments."
Hammel, who will be slotted into the third spot in the Cubs' rotation, gave up two runs on eight hits and three walks over 4 2/3 innings. He struck out two.
The Athletics took advantage of two errors by third baseman Mike Olt in the seventh to add three more runs. Olt made up for his miscues when he led off the eighth with his fifth home run.
Lake's third homer of the game came in the ninth, off Oakland reliever Fernando Nieve, who also allowed Valbuena's second of the game in the next at-bat.
Chicago has hit 33 home runs, which ranks among the Major League leaders this spring.
Up next: Minor League right-hander Murphy Smith will make his first Cactus League appearance when he starts against the Texas Rangers on Monday at 1:05 p.m. PT at Surprise Stadium, live on MLB.TV or through an exclusive athletics.com webcast. Smith pitched the past two seasons in Double-A and had a 3.48 ERA in 25 starts in 2013. The Rangers will counter with left-hander Ryan Feierabend.
Carrie Muskat is a reporter for MLB.com. She writes a blog, Muskat Ramblings, and you can follow her on Twitter @CarrieMuskat. This story was not subject to the approval of Major League Baseball or its clubs.