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Eduardo Rodriguez went from awful to awfully good in five days.
Rodriguez (3-1), who gave up nine runs in 4 2-3 innings against Toronto on Sunday, allowed one run and six hits in 6 1-3 innings.
"A back bounce start for him, there's no doubt," Red Sox manager John Farrell said. "He threw three quality pitches for strikes, but most importantly his fastball location was outstanding.
"We put up the five runs in the second and he pitches as if it was a nothing-nothing game. There's fluidity, concentration and quality execution. He backed up a rough outing a few days ago with a solid outing tonight."
Alcides Escobar's RBI-double in the fifth was the first run he allowed on the road in 18 1-3 innings.
"I mixed my pitches," Rodriguez said. "I threw to both sides of the plate with a slider and changeup."
The Royals could not solve the rookie left-hander.
"He was 96 with the fastball and 86 with the changeup," Royals manager Ned Yost said. "When you have an 8 to 10 mile-per-hour variation on your fastball and changeup and can command it, you're going to be tough."
The Red Sox batted around in the five-run second inning, which included an RBI-single by Betts. Blake Swihart contributed a two-run double, while Mike Napoli drove in two runs with a single.
Betts is 13 for 22, .590, in his past five games, raising his average to .267.
Boston's Hanley Ramirez hit his 14th home run in the fifth.
Yohan Pino (0-2), who was making his first start, was roughed up for six runs and 11 hits in 5 1-3 innings. Pino was recalled from Triple-A Omaha to fill in for Yordano Ventura, who went on the disabled list with ulnar nerve inflammation.
"You have to forget that (second) inning, keep working and make your pitches," Pino said.
The Red Sox' 16-hit attack featured five players with multi-hit games, including Pablo Sandoval, who doubled and singled to extend his hitting streak to nine games. Sandoval, who hit .429 against the Royals in the 2014 World Series, was benched Thursday after he used his cellphone during Wednesday's game in Atlanta.
The Red Sox have won seven of their past eight games against the Royals.
Eric Hosmer homered in the eighth for Kansas City, which added a run in the ninth on a double by Mike Moustakas.
Red Sox: 2B Dustin Pedroia (left knee inflammation) was not in the lineup for the third time in four games, but is expected to play Saturday.
Royals: RHP Kris Medlen (reconstructive elbow surgery) made his first rehab start Friday for Double-A Northwest Arkansas, going four innings and allowing three runs, one earned, and four hits, while striking out five, walking none and hitting two batters. He threw 39 strikes in 60 pitches.
Red Sox: RHP Rick Porcello, who is 0-5 with a 6.97 ERA in his past five outings, starts Saturday.
Royals: RHP Edinson Volquez, who starts the middle game of the series, is three strikeouts shy of 1,000.
The Red Sox challenged Swihart was safe when he was called out attempting to steal second in the fourth inning. After a 4-minute, 37-second delay, the ruling on the field was not overturned.
The Red Sox grounded into five double plays, including David Ortiz and Swihart twice each.
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