Kershaw (2-1) struck out seven, walked two and gave up six hits in 6 2/3 innings, and he hasn't allowed a run over his last 23 2/3 innings against the Giants.
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"We need a better game plan for him," Burrell said. "He's tough, no doubt about it. He's one of the tougher guys out there. He's got tremendous stuff and when he's locating his secondary pitches, he's even tougher."
Burrell broke up Los Angeles' shutout bid with a solo homer against Mike MacDougal leading off the ninth that hit the left-field foul pole. He had given the Dodgers a run in the fifth inning, allowing Matt Kemp's single to get past him for an error.
"I've made bad plays," he said. "We all have. There's no excuse. The tough plays are going to happen. The routine plays have to be caught every time."
Only the Houston Astros have a lower fielding percentage in the NL than the Giants.
"We know we need to tighten up this defense," Giants manager Bruce Bochy said. "We've let it get away from us and it's hurt us."
The game was dedicated to longtime Giants fan Bryan Stow, who was severely beaten by two men in Los Angeles gear outside Dodger Stadium after the teams' March 31 season opener. Stow, 42, is at a Los Angeles hospital in critical condition and in a medically induced coma.
The Giants heightened security at AT&T Park in the wake of what happened - as is standard practice when the Dodgers come to town anyway.
After a jam-packed weekend of World Series celebrations, the Giants presented former infielder Juan Uribe with his championship ring before the game. Then, the tone became far more serious.
On a night when players from both teams addressed fans before first pitch, Giants reliever Jeremy Affeldt and Dodgers second baseman Jamey Carroll came together for a joint message to the sellout crowd of 40,870: This rivalry must stay on the field, without violence and hatred.
The first seven batters reached base against Madison Bumgarner (0-2) in the fifth, which resulted in four runs.
"It was me not being able to put guys away," Bumgarner said. "It was a frustrating game. A lot of things didn't go my way."
Aubrey Huff had two hits for the Giants, who were coming off their first series victory of 2011 against St. Louis over the weekend. San Francisco lost two of three to open the season at Dodger Stadium, both by one run. The Giants have been held to one or fewer runs in four of their six losses.
Bumgarner, the 21-year-old lefty who shut out the Texas Rangers in Game 4 of the World Series as a rookie, was tagged for five runs on eight hits in five innings.
Last season, he went 6-3 with a 1.91 ERA in 10 outings away from the Giants' waterfront ballpark last season and 1-3 with a 4.60 ERA in eight starts at home.
- Kershaw lowered his ERA from 2.08 to 1.37.
- Kemp stole his seventh base for the Dodgers in the second inning after drawing a leadoff walk.
- The Giants plan to decide Tuesday whether injured CF Andres Torres, who strained his left Achilles' tendon Saturday night, will need a disabled list stint. They don't want to go short-handed for long. Miguel Tejada batted leadoff in Torres' usual spot Monday.
- Injured Giants RF Cody Ross has resumed jogging and outfield work as he recovers from a strained right calf that landed him on the DL to start the year. He gave a thumbs-up sign after the workout. Ross is slated to get some at-bats in extended spring training when the team heads to Arizona this weekend, then play in a couple of minor league games for a short rehab assignment.
- Bumgarner faced the Dodgers for the first time at home.