The game started out with the mercury at 103 degrees for the 5 p.m. start, setting a temperature record for the World Series.
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And the Dodgers bats started out hot, too.
Chris Taylor swung at the first pitch in the bottom of the first and knocked it clear of the Dodger Stadium left-field fences to start LA off with a 1-0 lead.
But then it took some time for the Dodgers to build on the momentum.
First, three innings later the Astros answered with their own solo homer off Kershaw from Alex Bregman.
The game stayed at a 1-1 tie until Justin Turner hit a two-run homer in the sixth, to make it 3-1.
With dominating pitching from Clayton Kershaw, and relief from Justin Morrow and Kenley Jansen, that score held through the end.
Kershaw was starting in his first World Series and put up historic numbers. He threw 11 strikeouts though 7 innings, giving up only 1 run.
That made him the first Dodgers pitcher to strike out 10 batters in a World Series game since it was done in 1965 by Sandy Koufax, who was in Dodger Stadium to watch Kershaw work.
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Kershaw struck out the side twice during the first six innings, and his off-speed pitches looked particularly vicious on a sweltering night. He overwhelmed the Astros, who struck out fewer times than any big-league team - and never struck out 11 times in any game - during the regular season.
Morrow replaced him in the 8th, with Jansen brought in at the end for the save.
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A few other familiar names to Dodger fans were also in the stands.
Vin Scully showed up, but despite pleas from fans he attended as a spectator, not a broadcaster.
The 89-year-old Hall of Fame broadcaster who retired at the end of last season was shown on the video board in between innings, drawing big cheers from the crowd of 54,253. He was sitting in a suite between Jackie Robinson's widow Rachel and his wife Sandi.
Scully smiled and waved at the crowd and put his hands together in a gesture of thanks.
Scully was the longest-tenured broadcaster with the same team in pro sports when he retired. He spent his entire career with the Dodgers, having first joined the team when it was located in Brooklyn, where he became friends with Robinson, who broke baseball's color barrier 70 years ago.
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Jackie Robinson's family threw out the first pitch before the game.
Robinson broke baseball's color barrier 70 years ago with the Dodgers.
His children Sharon and David Robinson stood in front of the mound to throw pitches. They were joined by Robinson's 95-year-old widow, Rachel, who climbed out of a golf cart to throw the ball a short distance.
Robinson's number 42 is retired around the major leagues.
Outfielder Joc Pederson's brother Champ shouted "It's time for Dodger baseball." He has Down's syndrome.
Wearing a Clayton Kershaw jersey, comedian George Lopez stood atop the Dodgers' dugout and waved a blue-and-white team flag.
Game 2 starts at 5:09 p.m. Wednesday at Dodger Stadium.
The Associated Press contributed to this report.
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