The Cubs had a five-run rally in the fifth inning to take a 5-2 lead but the Brewers came back with a big rally of their own two innings later to beat James Russell (1-3).
Russell entered the game with one out in the seventh and runners on first and second. Davis connected with Russell's first pitch for his second home run of his major-league career. After another out, Norichika Aoki singled and stole second and third ahead of a Jean Segura double that put Milwaukee ahead.
Rob Wooten (1-0) got his first major-league win in relief but Brewers closer Jim Henderson made it interesting with two ninth-inning walks before finishing off his 13th save.
The Brewers led 2-0 after a fourth-inning home run by Segura and a fifth-inning homer from Yuniesky Betancourt. It was the 12th home run of the season for each.
Chicago bunched together six singles and a walk in the fifth for all five of its runs. Darwin Bailey drove in the first two runs and Wellington Castillo, Carlos Villanueva and David DeJesus followed with consecutive RBI base hits for a 5-2 lead.
Yosemite climber falls 30 feet, suffers major injuries
'How to Train Your Dragon 2' releases 5-minute clip