Can the rotation-starved Royals allow veteran right-hander Jeremy Guthrie to depart in the off-season as a free agent? Barring a mind-blowing offer from another team, how can they?

Guthrie closed his season Tuesday night with six strong innings in a 4-2 victory over the Detroit Tigers at Kauffman Stadium. He recorded his 10th quality start in his final 12 outings by yielding two runs and five hits.

“Great defense behind me,” he said. “Not just infield defense, but also outfield defense, which gets overlooked a lot. The guys run and get it, cut off doubles, and throw runners out.

“It’s a place where any pitcher is going to be made a whole lot better because of the guys around him on the diamond.”

Both Detroit runs came on a single by Triple Crown candidate Miguel Cabrera in the third inning. The Royals backed Guthrie with three shutout innings from the bullpen, home runs by Alcides Escobar and Jeff Francoeur. And a Gold Glove resume night from Escobar at shortstop. His play deep in the hole, and jump throw, on Avasail Garcia’s one-out grounder in the eighth inning was special even by Escobar’s standards.

“That was a great game for me,” Escobar said, “and I’m so happy because it was on TV back home in Venezuela. I think everybody saw the game. I’ve already got 35 messages on my phone.”

“The best part? Everything. The homer. The RBI (single in the fifth). The play (in the eighth). And we won the game.”

But Escobar is under club control through 2017. The more pressing issue is Guthrie, a pending free agent who was 5-3 with a 3.16 ERA in 14 starts following the July 20 trade that brought him from Colorado for Jonathan Sánchez.

What comes next?

“The people who handle those negotiations are going to work on those,” Guthrie said. “I know they’ll do the best they can, and we’ll see where it goes from there. You take care of your family first.

“You figure out what the best situation is overall, both for you and your family, and you go from there.”

Guthrie said he sees the Royals as a good potential long-term fit.

“Oh, absolutely,” he said. “I think so. Great teammates. Great city. I think any pitcher would be fortunate to pitch here.”

Hey, maybe Guthrie can play salesman. The Royals figure to be looking for two or three starting pitchers, maybe more, in the coming offseason.

Guthrie handed a 3-2 lead in the seventh to Francisley Bueno, who worked around a one-out single and an infield misplay for a scoreless inning. Kelvin Herrera pitched a one-two-three eighth with two strikeouts sandwiched around Escobar’s marvelous play.

Escobar then produced an encore for the first out in the ninth on Bryan Holaday’s grounder. That benefited Greg Holland, who closed it out from there for his 16th save in 18 chances since becoming the club’s closer.

The victory means the Royals, at 72-89, will finish with a better record than last year’s 71-91 mark. It also slowed a late-season slide of eight losses in the previous nine games.

“They don’t quit,” manager Ned Yost said. “They go until the last game’s played, until the last pitch is thrown. That’s why you can sit back and feel that our future is so bright. We have kids with tremendous makeup that continue to fight.”

It was 3-2 when Francoeur opened the eighth by crushing a first-pitch fastball from reliever Brayan Villarreal, who had just entered the game. The ball traveled 424 feet to the left of dead center for a two-run lead.

Cabrera aided his Triple Crown bid by going two for three before exiting in the fifth inning. He leads all three categories: 44 homers, 139 RBIs and a .331 average with one game remaining in his bid to be the first to achieve the feat since 1967.

Detroit manager Jim Leyland said Cabrera will play tonight if he wants to. Cabrera’s response: “Why not?”

The Royals grabbed a 1-0 lead in the first inning when Escobar, who finished with three hits, jumped a 2-2 fastball from Tigers starter Doug Fister, 10-10, for a one-out homer. The 378-foot drive went into the Royals’ bullpen beyond the left-field wall.

Detroit nicked Guthrie for two runs in the third after a one-out walk to Danny Worth. Singles by Quintin Berry and Ramon Santiago loaded the bases for Cabrera, who pumped a two-run single into center for a 2-1 lead.

A walk to Prince Fielder reloaded the bases with one out before Guthrie caught a break. Fielder broke for second on Delmon Young’s infield flare that Tony Abreu easily ran down at second. Abreu flipped to first baseman Billy Butler for the double play.

A between-innings discussion with pitching coach Dave Eiland produced a mechanical adjustment that enabled Guthrie to find a groove. He permitted just one hit in his final three innings.

“I wasn’t getting my hand out,” he said, “and I was way up in the zone to the point where I wasn’t able to get it back down for those two or three hitters. After I got out of that (inning), and figured it out, I was able to settle back in.”

Cabrera exited after fouling out to right for the final out in the fifth.

The Royals scored twice later in the inning when the Tigers’ suspect infield defense got fully exposed. Irving Falu led off with a single past a diving Fielder at first. Falu went to third when Jarrod Dyson singled past a diving Santiago at second.

Both grounders appeared catchable.

Escobar tied the game with a single through the left-side hole — a clean single. Alex Gordon popped to short before Fister loaded the bases by walking Butler. Leyland went to the bullpen for Luke Putkonen to face Salvy Perez.

That should have worked.

Perez hit what should have been a double-play grounder to short, but Santiago double-clutched on the relay, which wasn’t in time to beat Perez at first. Dyson scored, and the Royals led 3-2.