Kershaw (3-1) struck out eight while working six innings of three-hit ball. He also drew a bases-loaded walk against Yu Darvish (1-1) in the fifth, leading Los Angeles to its eighth straight win.
“We knew that this was going to be an emotionally intense (series) obviously coming here,” Dodgers manager Dave Roberts said. “It certainly lived up to the billing.”
Even with Kershaw’s stellar performance, the matchup wasn’t decided until Betts turned in another memorable play for the final out.
With runners on second and third, Tommy Pham hit a sinking liner to center that looked as if it was going to tie the game. But Betts got over for a terrific diving grab, and then pounded on his chest in celebration.
“It takes an elite defender in center to make that play. A lot of things have got to line up,” Roberts said. “There are just a handful of guys that probably could have made that play. It just speaks to that is why he is arguably the best all-around player in baseball.”
Justin Turner, who homered in the ninth for Los Angeles, said Betts saved the game for the World Series champions.
“He is a guy who can change the game and impact the game in so many different ways,” Turner said.
Said Kershaw: “That play is really hard to make. For him to be able to do that is just another reason why Mookie is the best.”
Los Angeles also took the series opener Friday night, grabbing the lead for good when Corey Seager hit a tiebreaking two-run homer on Tim Hill’s first pitch of the 12th inning.
The second game of the season series was just as good.
Kershaw won his third straight start since losing at Colorado on opening day. Victor Gonzalez got two outs for his first save of the season.
Darvish was terrific for San Diego, allowing one hit in seven innings. He struck out nine and walked two.
Los Angeles grabbed the lead with a two-out rally in the fifth. Zach McKinstry was hit by a pitch, Luke Raley singled and Austin Barnes walked to load the bases.
Kershaw then walked on a full-pitch count, forcing in a run. Darvish’s 86-mph cutter was slightly low and outside.
San Diego had runners on the corners with one out in the sixth when Wil Myers hit a sharp grounder up the middle. Second baseman Chris Taylor grabbed the ball, stepped on second and threw to first baseman Max Muncy for a double play to end the inning.
Turner gave the Dodgers an insurance run with a solo drive off Emilio Pagan in the ninth.
Dodgers: Infielder Gavin Lux was available to pinch run, but he wasn’t allowed to hit due to a sore right wrist.
Padres: Placed RHP Dan Altavilla on the 10-day injured list with right elbow inflammation. LHP Matt Strahm (right patellar tendon repair) was transferred to the 60-day IL. … 3B Manny Machado started despite jamming a shoulder sliding into second base and having back spasms on Friday night.
Dodgers right-hander Trevor Bauer (2-0, 2.70 ERA) and Padres left-hander Blake Snell (1-0, 3.06 ERA) are slated to pitch Sunday.