Bumgarner struck out 10 while working into the seventh inning, and the Giants got home runs from Michael Morse, Pablo Sandoval and Buster Posey in a 7-2 victory on Sunday.
"We've always had a good defensive team, and we're starting to swing the bats now," Bumgarner said. "So we're a pretty well-rounded team. Today the bullpen came in and did a great job, and the offense was unbelievable again. It's nice to have that kind of support. It makes it easier to go after guys and challenge them."
Right fielder Yasiel Puig returned to the Dodgers' starting lineup, batting leadoff a day after he was benched by manager Don Mattingly for arriving late for stretching and batting practice before Friday's home opener. Puig went 1 for 4 and was picked off first in the third inning for the final out with cleanup-hitter Adrian Gonzalez at the plate.
Bumgarner (1-0) was charged with two runs and eight hits over 6 1-3 innings in his second outing.
"I was throwing a little bit of everything," the left-hander said. "It seemed like they had a lot of baserunners today, but I felt like I made good pitches all day. The command was there and my stuff was pretty good."
Bumgarner has a career strikeout-to-walk ratio of 3.65, which is better than that of All-Star teammates Tim Lincecum (2.77) and Matt Cain (2.48). Last season, he came within one strikeout of becoming only the second Giants lefty since Ray Sadecki in 1968 to finish a season with 200 or more.
"You don't want to give guys free passes at any time. You have to make them earn it of you can," Bumgarner said. "But at the same time, you don't want to just give in and just throw a meatball up there. You've got to make pitches and work on command and get ahead of guys."
Bumgarner is 4-1 with a 1.53 ERA in his last five starts at Chavez Ravine.
"He was commanding both sides of the plate and locating," Gonzalez said. "He's just like anybody else. If he can command down and away, down and in, up and away and up and in, and crisscross the strike zone, it makes it tough."
Puig came up in the seventh with the bases loaded and one out after Giants skipper Bruce Bochy replaced Bumgarner with Santiago Casilla. Puig's flyball to right field was too shallow to score Scott Van Slyke, but pinch-hitter Andre Ethier followed with an RBI single.
A.J. Ellis tried to score behind Van Slyke and was thrown out by Hunter Pence for the third out. Mattingly came out to discuss the call with umpire Joe West and used his one challenge for a video replay ruling, which upheld the original call.
"I had to make a pretty good throw and Buster had to make a great tag," Pence said.
Paul Maholm (0-1) gave up five runs and seven hits through 4 1-3 innings in his first start for the Dodgers.
"Maholm's got a lot of different pitches that he throws it in a lot of different ways, and he located really well. So it's always tough," Pence said. "You just battle, make him get the ball over the plate and be aggressive."
Morse, playing for his fourth team in a span of three seasons, gave San Francisco a 2-1 lead when he drove Maholm's first pitch of the fourth deep into the pavilion seats in left-center for his first homer with the Giants.
Sandoval, the Giants' slimmed-down third baseman and 2012 World Series MVP, parked Maholm's 80th and final pitch over the left field fence in the fifth for with two on for his first home run of 2014. Posey then greeted rookie Jose Dominguez with a drive into the lower seats in the left field corner.
NOTES: Maholm had allowed only three homers in 62 innings during his 10 previous career starts against the Giants, all of them solo shots. ... Maholm's official Dodgers debut was March 23 against the Diamondbacks in Australia, when he pitched in relief for only second time in 244 big league appearances to that point. ... Angel Pagan worked out an eight-pitch walk to open the game and was erased on a double-play grounder by Pence, who was robbed of a hit up the middle by shortstop Hanley Ramirez. Maholm has induced a major league-high 185 ground-ball DPs since the start of the 2006 season.