Cain's RBI groundout three batters in put the Royals ahead, and Kansas City expanded its lead to 3-0 in the sixth when Alex Gordon hit an RBI double off Cain and scored on Eric Hosmer's single off left-hander Javier Lopez.
San Francisco finally got to Guthrie in the bottom of the sixth when pinch-hitter Michael Morse hit an RBI double over third base just past the glove of a diving Michael Moustakas and scored on Buster Posey's groundout against Kelvin Herrera.
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Herrera, Brandon Finnegan, Wade Davis and Greg Holland combined for four innings of hitless relief, the longest in a Series game in 22 years.
Game 4 is Saturday night with Kansas City's Jason Vargas facing San Francisco's Ryan Vogelsong.
Of the 57 times a World Series has been tied 1-1, the Game 3 winner has taken the title 37 times.
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Guthrie returned to the area where he gained attention pitching for Stanford in 2001-02, becoming a first-round draft pick, and made just his second start since Sept. 26. He allowed two runs and four hits in five-plus innings.
Guthrie gave up an infield single in the second to Hunter Pence, who was caught stealing second by catcher Salvador Perez, followed by Brandon Belt's single to center. Guthrie retired his next 10 batters in order and combined with Hudson to get out 20 in a row, the longest Series streak since the Yankees' Don Larsen and the Brooklyn Dodgers' Sal Maglie retired the first 23 batters during Larsen's perfect game in 1956, according to STATS.
At 39 years, 102 days, Hudson became the second-oldest pitcher to make his World Series debut as a starter behind Philadelphia's Jamie Moyer. Moyer was 45 years, 342 when he started the third game against Tampa Bay in 2008. He allowed three runs and four hits in 5 2-3 innings.
On a sun-splashed late afternoon in Northern California, the Fall Classic returned to AT&T Park, where the Giants had won six straight Series games dating to 2002. The ballpark, alongside San Francisco Bay, has a unique atmosphere: The Grateful Dead's "Sugar Magnolia" and "Uncle John's Band" played on the sound system during batting practice, and public address announcer Renel Brooks-Moon stirred up the crowd with an excited voice. Many fans wore orange - including a man in an orange suit behind home plate - and thousands wore orange towels.
With the shift to the NL ballpark, the Royals moved Cain from center field to right to boost defense in one of AT&T Park's trickiest positions. Usual right fielder Nori Aoki was not in the starting lineup, and Jarrod Dyson was in center and hit eighth in his first start since Sept. 20.
Cain made a pair of nice plays early, sliding to grab Posey's liner to end the first and reaching down for a running grab on Travis Ishikawa's twisting drive with a runner on to end the second.
Gordon, who batted sixth in Kansas City, moved up to No. 2 hole as the Royals lost their designated hitter, Billy Butler. And with the loss of the DH, San Francisco's Michael Morse was out.
Hall of Famer Tony Gwynn, who died June 16 of oral cancer, was honored before the game. His family stood on the infield with MLB Chief Operating Officer Rob Manfred as a tribute was played on the center-field video board that included Gwynn highlights and interviews. The ceremonial first pitch was thrown out by Frank Burke, the fan who caught Ishikawa's pennant-winning home run last week and gave the ball to the Giants outfielder.
There also was a video remembrance of the 1989 earthquake that happened just before the scheduled start of Game 3 at Candlestick Park, and rock'n'roller Huey Lewis shouted "Play ball!"
Alcides Escobar drove Hudson's first pitch, a 91 mph fastball, off the base of the wall near the left-field corner, advanced when Gordon grounded the next pitch to first and scored on Cain's bouncer to shortstop.
Hudson, in his 16th big league season, got in trouble again in the second, when Mike Moustakas singled to lead off and Omar Infante walked. Perez lined a hanging cutter to left, when Ishikawa made a sliding catch. Dyson, among the fastest runners in the major leagues, grounded into a 6-4-3 double play.
Escobar grounded a single up the middle past Hudson with one out in the sixth, ending Hudson's streak of 12 batters retired. Gordon drove the next pitch on a hop to the wall in center, ending an 0-for-9 Series slump with his 10th RBI of the postseason.
After Cain grounded out, Hosmer fouled off six pitches before singling to center on the 11th pitch of the at-bat, ending his 0-for-8 Series slide.
Brandon Crawford singled leading off the bottom half, and Morse pulled a deep drive about 15 feet foul before doubling.
Herrera, throwing at up to 101 mph, walked Gregor Blanco and then induced consecutive groundouts from Joe Panic, Posey and Pablo Sandoval.