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: Carlos Villanueva, who could fill the fifth-starter spot at the beginning of the season, will make his second Cactus League start Monday night when the Cubs play host to the Padres at Cubs Park. Monday's projected lineup will have Emilio Bonifacio at the top of the order. It will be the first night game at the Cubs' new facility, which is averaging 14,020 fans. The game is available on Gameday Audio (WGN Radio).