Bumgarner (3-0) earned his third win in three starts to begin the season, shutting down the Cubs as San Francisco built a 3-0 lead.
The left-hander kept the Cubs off the board until until surrendering a two-out, two-run homer to Dioner Navarro in the seventh inning. He finished allowing six hits and two walks while striking out six.
Jeremy Affeldt finished up the seventh for Bumgarner and Santiago Casilla hurled two scoreless frames to pick up his first save of the season as the Giants rebounded from Friday's 4-3 loss at Wrigley Field, where the game-time temperature Saturday was 38 degrees.
Chicago's Jeff Samardzija (1-2) absorbed the loss, yielding a pair of runs on seven hits over six innings.