In his first appearance since his 15-strikeout no-hitter against the Rockies in which the only man that reached base did so on an error, Kershaw (8-2) struck out eight and walked one while giving up six hits in eight innings for his fifth win in as many starts.
Danny Duffy (4-7) dropped his second straight start despite giving up just one run on four hits over six innings for the Royals, who have lost five of six.
The Dodgers jumped on top in the first inning as Justin Turner tripled to start the game. He was held there when Matt Kemp grounded out and Yasiel Puig reached on an infield single, but Gonzalez got him home on a fielder's choice for a 1-0 lead.
Hosmer again singled with one out in the fourth and moved to second on a wild pitch with Alex Gordon at the plate. Gordon eventually worked a walk, but Salvador Perez grounded out to end the inning.
Kershaw tossed a 1-2-3 eighth inning, and the Dodgers gave themselves some insurance in the ninth.
Gonzalez walked to start the inning and moved to second on an A.J. Ellis single. Ethier then delivered a pinch-hit single for a 2-0 lead.
Kenley Jansen retired the Royals in order in the ninth for his 23rd save of the season.