Kershaw (1-0) and the Giants' Matt Cain engaged in a scoreless pitching duel until the Dodgers left-hander led off the eighth inning with a solo shot against reliever George Kontos (0-1) for his first major league home run.
Andre Ethier and A.J. Ellis went on to add RBI in the decisive inning as Los Angeles built a 4-0 lead.
Kershaw then finished up his sixth career shutout. The 2011 Cy Young Award winner didn't allow a walk and struck out seven in his third straight Opening Day assignment for the Dodgers.
Cain struck out eight over six scoreless frames for the defending champion Giants, allowing four hits and a walk in the no-decision.