Home runs have been a problem for Kennedy, who came into the game having allowed 27 this season, the second-most in the National League. He threw his arms above his head when he realized that Kubel had caught the ball.
"You don't expect those plays," Kennedy said. "He had really good timing and really fooled all of us. It definitely brought our spirits up seeing something like that, because I made a bad pitch on that pitch. I completely missed. I was trying to go down and away and it was kind of up, middle. To see a play like that, it was a big break."
Kennedy (13-11) and the D-backs needed all the breaks they could get considering they were facing Dodgers ace Clayton Kershaw.
Kershaw (12-9) held the D-backs to three hits over his seven innings, but the left-hander walked three and had to pitch out of a bases-loaded jam in the third.
With the game still scoreless, Paul Goldschmidt led off the seventh with a grounder to short, and Hanley Ramirez was charged with an error when his throw to first was in the dirt.
"We took advantage of their one mistake, the Ramirez error, that's why you hustle down the line," D-backs manager Kirk Gibson said. "Hanley had to back up on it and he wasn't able to step through it because Goldy was running hard."
One out later, Miguel Montero lined a double to right-center that scored Goldschmidt with the unearned run.
"I blinked first tonight," Kershaw said. "I threw a pitch I shouldn't have. I didn't put it where I wanted to and Miguel made me pay for it. He did a good job with it. You can't blink first in those games. I did, and that's where it goes. I was ahead in the count, I wanted to get him to chase one, but I left it over the plate"
The D-backs missed an opportunity to increase their lead that inning when Montero, who had moved to third on a passed ball, was thrown out at the plate by Matt Kemp while trying to score on Chris Johnson's flyout to center.
Kennedy wound up allowing just four hits in 7 1/3 innings. He did not walk a batter while striking out six.
"I thought Ian was really good early, he kept that fastball down and his changeup was good," Dodgers manager Don Mattingly said.
A key to Kennedy's success was his ability to throw his offspeed pitches for strikes. That allowed him to not have to be too fine with his fastball.
"You've got good hitters over there," Kennedy said. "It's a really, really tough lineup and you've got to keep them off balance. You're going against Kershaw and he's one of the best pitchers in the game."
Kennedy is plenty familiar with Kershaw. The pair have faced off five times in their careers, with Kennedy now holding a 3-2 edge.
"He's one of the best, and it's always fun pitching against him," Kennedy said before smiling. "It's really hard hitting against him. That's why when I found out he was starting tonight I was like, 'Gosh dang, I thought I was going to get a break hitting off him.'"
The win was the first 1-0 victory by the D-backs this year and the third time they've beaten Kershaw. Kershaw did get a win against them in May, but dropped the last three.
"We're going to play hard every game until the season is over," Gibson said. "We want to be as good as we can be, and let's find out where it is."