"Aaron's facing the best hitter to walk on this planet," Royals first baseman Eric Hosmer said. "The way he came in and showed no fear, went right after him with the crowd standing up on every pitch. That's a big moment right there. Crow is usually not a big fan of showing emotion right there, but he got fired up. It just fires up the whole team at that point."
Guthrie (7-3) limited the Tigers to a two-run homer by Cabrera in 6 1/3 innings in helping the Royals to extend their longest winning streak since they won seven straight in September 2011. He gave up six hits and walked three.
Crow replaced Guthrie after he gave up a single to Avisail Garcia and he hit Andy Dirks with an 0-2 pitch. Crow retired Torii Hunter on a fly to shallow right and struck out Cabrera swinging to end the threat.
"The slider was probably the best pitch I've thrown all year," Crow said of the pitch he used to strike out Cabrera. "I haven't thrown that many good ones. That's what we're paid to do. It's a situation you don't want to be in."
The Tigers managed just one two-out single off four Kansas City relievers after Guthrie exited with Crow logging the biggest two outs.
"The bullpen was huge for us," Guthrie said. "They won the game for us."
Cabrera, who tops the American League with 90 hits and 69 RBIs, hit his 18th home run in the second after Hunter doubled.
"I made a bad mistake and he didn't miss it," Guthrie said of hanging a breaking ball.
Perez's one-out triple in the third by diving center fielder Garcia scored Alcides Escobar and Hosmer after they each singled. Lorenzo Cain's two-out infield single scored Perez with the go-ahead run.
"Garcia gave a good effort and missed," Hunter said. "I chased it down. As an ex-center fielder, I know, if you see me coming, you get out of the way. That's his ball if he wants it. I've got to get out of the way. On that ball, a right-handed batter, the ball fades back to right field. It's probably coming back towards me. He probably should have let it go. He played it aggressively and that's all right. He's 21. He'll learn. He's 6-5, 250. I'm getting out of his way."
Doug Fister (5-4), who is 0-3 in five starts since a victory May 14 over Houston, went the distance. He allowed three runs on nine hits.
"They're patient," Fister said. "When they needed to they produced what they needed. They put two runs on the board for me and I didn't make it stand up."
Greg Holland worked a flawless ninth for his 13th save in 15 opportunities.
"Just a phenomenal job of pitching on our part," Royals manager Ned Yost said.
The Royals are 7-2 in June after losing 20 of 28 games in May. The pitching staff has a 1.44 ERA in June.
The Royals improved to 8-3 since George Brett was named the interim hitting coach on May 30.
Tigers CF Austin Jackson, who went on the disabled list May 12 with a left hamstring pull, began an injury rehab assignment Monday with Triple-A Toledo. He went 0-for-5, struck out three times and walked once at Rochester. ... Royals LHP Danny Duffy, who had Tommy John surgery last June, made a rehab start Monday for Class-A Omaha, giving up seven runs on seven hits, two walks and a hit batter in 2 1/3 innings, throwing 58 pitches at Nashville. ... The Royals signed Stephen F. Austin SS Hunter Dozier, the eighth overall pick in the draft, to a $2.2 million signing bonus, which is $937,800 under the allotted slot bonus. ... RHP Anibal Sanchez, who missed a start Sunday with shoulder stiffness, will probably make his next start. "I think so," Tigers manager Jim Leyland said. ... The Tigers recalled RHP Evan Reed from Triple-A Toledo, where he was 1-1 with a 2.53 ERA in 21 outings.