Kershaw received 27 of 32 first-place votes, five seconds and two thirds for 207 points in voting by the Baseball Writers' Association of America.
"I always dreamed about playing in the big leagues. I never dreamed about doing anything special in the big leagues. I don't think any kid ever does," Kershaw said. "The people I'm now associated with, just by having this award, is something that I never thought would ever happen."
Philadelphia's Roy Halladay, last year's winner, was second with four first-place votes, 21 seconds and seven thirds for 133 points. Phillies teammate Cliff Lee was third with 90 points, followed by Arizona's Ian Kennedy with 76 points.
"Whenever you have a Cy Young next to your name, there's going to be expectations that go along with it," Kershaw said. "Whenever I look at a pitcher and I see that he's won a Cy Young Award, I think, you know, this guy, he better be good. And that's what I hope to be. I hope people have that expectation for me."
With a big curveball that might be the best in baseball, Kershaw won the NL's pitching triple crown. Pitching on a team that went 82-79, he led the league with a 2.28 ERA and 248 strikeouts and with a 21-5 record tied Kennedy for most wins.
The 23-year-old left-hander, whose previous high for victories was 13 in 2010, dominated the league during the final two months of the season, going 8-0 with a 0.96 ERA in his final nine starts.
His .207 opponents' batting average was the second-lowest in the major leagues among qualified pitchers, trailing only Detroit's Justin Verlander (.192), the unanimous AL Cy Young winner. Kershaw was especially effective at Dodger Stadium, where he went 12-1 with the lowest home ERA in the big leagues at 1.69.
Los Angeles has a chance to sweep the two major NL awards. Matt Kemp is a favorite to win the MVP, which is announced Tuesday.
It was the 10th Cy Young won by the Dodgers, following Don Newcombe (1956), Don Drysdale (1962), Sandy Koufax (1963, 1965 and 1966), Mike Marshall (1974), Fernando Valenzuela (1981), Orel Hershiser (1988) and Eric Gagne (2003).
Kershaw downplayed any comparisons with Koufax.
"I'm still uncomfortable with it. I don't want to have any disrespect for Mr. Koufax. He did it for a long time. He won a lot of awards and he won World Series. He threw no-hitters. Just a lot of things I'm not anywhere close to accomplishing yet," Kershaw said. "I have tremendous respect for him and would never want to ever put myself in the same category as him."
A bargain with a $500,000 salary, Kershaw did not have a bonus provision. Halladay won $75,000 for finishing second and Lee $50,000 for finishing third.
A Dallas native who still lives in Texas, Kershaw said he comes from an area where "football is king."
Still, if kids want to emulate him, they can play baseball in the springtime.
"Anything to stay away from lacrosse, in my opinion," he said with a laugh.