Guthrie didn't make much progress in that endeavor on Sunday.
Guthrie (1-2) allowed all six runs on 11 hits in six innings, striking out just one. He has been pounded for at least four runs in four of his five starts this season, and gave up all the runs in a 3-0 loss to Minnesota the only time he fared any better.
"It's the third day of May," Yost said. "These guys work really hard. When they're going through periods like this, they continue to work hard. They'll figure it out."
Kansas City nearly bailed out Guthrie by scoring two runs in the seventh and two more in the eighth. But Tom Gorzelanny struck out Eric Hosmer to leave a runner aboard, and Joakim Soria shut down his former team in the ninth for his 10th save.
"I felt much, much better than I have in a couple of starts," Guthrie said. "I wish it could have translated into getting more outs and being more effective."
The Tigers dropped the first two games of the series before squeaking out a 2-1 win behind David Price's masterpiece Saturday night. Anibal Sanchez (2-3) followed that by carrying a perfect game into the sixth inning Sunday, and allowing three runs on four hits in 7 1-3 innings.
"That's the big key, when you throw your first pitch for strikes," Sanchez said. "They're pretty hot right now, so that's why I had to keep the ball down."
Guthrie struggled from the start, giving up an RBI single to Avila in the second inning and loading the bases for him in the fourth. Avila followed with a two-run single that cleared the bases when Paulo Orlando fumbled the ball.
Ian Kinsler added a single in the fifth before Cabrera hit his sixth homer this year.
Guthrie ended up allowing all six runs and 11 hits in six innings. The poor outing came after a dazzling start to the series by Kansas City pitchers, who had allowed Detroit's potent offense to score just four runs total in the first three games.
It would have taken a similarly strong outing to hang with Sanchez, who has struggled this season — the Tigers had lost his last four starts — but has owned the Royals. He'd been 5-2 with a 1.07 ERA in his previous seven starts against the reigning American League champions.
With a crowd of 38,326 packed inside Kauffman Stadium, the third straight sellout, the right-hander calmly mowed through the Royals lineup for most of the afternoon.
Kansas City did not have its second runner until Sanchez walked Kendrys Morales with one out in the seventh. The Royals went on to score on Erik Kratz's sacrifice fly and Omar Infante's single, then added runs in the eighth on a single by Morales and Mike Moustakas's groundout.
"We know the Royals pretty well. They know us pretty well," Tigers manager Brad Ausmus said, when asked to sum up the first series of the season between AL Central contenders. "I don't think there was any eureka moments. Just two pretty good teams."
The Royals welcomed 143,609 fans, the seventh-best attendance for a four-game set at Kauffman Stadium. It was a record for a four-game series since the 2009 renovation decreased capacity.
Royals: All-Star closer Greg Holland (strained pectoral muscle) is expected to come off the disabled list Tuesday. He reported no problems after a simulated game Saturday.
Tigers: RHP Shane Greene will take the mound after a day off to open a three-game set against the White Sox. The series beginning Tuesday night wraps up a 10-game trip.
Royals: LHP Jason Vargas tries to solve his early season struggles in the opener of a three-game series against Cleveland on Tuesday night. Kansas City is off Monday.
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