Kershaw struck out 11 and walked one in a 115-pitch effort in winning his eighth consecutive start.
"He was tough all night," Headley said. "He was throwing all of his pitches for strikes. He was ahead in the count all night, so you really couldn't wait him out.
"He showed why he's one of the best pitchers in the game."
The other guy -- Padres starter Odrisamer Despaigne -- wasn't so bad, either.
Despaigne couldn't match Kershaw (11-2) in terms of firepower, though the Cuban right-handed rookie -- armed with varying pitches, velocities and arm angles -- worked his way through the Dodgers' lineup just the same.
Despaigne (2-1) allowed two runs on seven hits in seven innings. He didn't walk a batter and also had a career-high seven strikeouts while getting 10 ground-ball outs.
After four big league starts, Despaigne has a 1.35 ERA in 26 2/3 innings.
"He showed us a lot," said Dodgers catcher A.J. Ellis. "That's an amazing arm over there. [He's] very crafty. It's just any pitch at any time and just an ability to change speeds and hit all corners. He never missed in the middle of the plate it felt like the entire night, at least in my at-bats"
Kershaw used more conventional methods -- his overpowering fastball but also a sharp breaking ball -- to give the Padres (40-52) fits.
"It seems like he's throwing his breaking pitches more," Headley said. "Last year, he was more fastball and slider."
Headley got a slider that didn't slide nearly enough with two outs in the sixth inning, sending it over the fence in left-center field for his seventh home run of the season, snapping Kershaw's streak.
"It stayed out over the plate and I was able to put a good swing on it." Headley said.
But good swings were few and far between for the Padres.
The Padres' defense was porous, as they tied their season high with four errors and nearly had a fifth when Adrian Gonzalez was credited with an infield single when shortstop Alex Amarista booted a ball in the eighth inning. The Padres also had four errors on April 26 against the Nationals.
The Padres played the last seven innings without first baseman Tommy Medica, who left the game after his first at-bat with tightness in his right groin. Catcher Rene Rivera came off the bench to play first base for the second time in his career -- the other time was also this season.
Medica will be re-evaluated Friday.
"That was uncharacteristic of how we've been playing," Padres manager Bud Black said. "Rene is out of position, so you can blame me. I felt we needed some offense out there."