A madcap sequence put the Twins ahead and a crazy play ended it, too.
With two outs and none on in the ninth, Mike Moustakas popped a ball straight up. As he stood to watch it in the batter's box, catcher Kurt Suzuki tripped over him, resulting in an interference call for the final out.
"That's always on the hitter," Twins manager Ron Gardenhire said. "That's not going to be on the defender. He has a right to catch the ball."
After the Royals scored three times in the eighth for a 3-2 lead, reliever Aaron Crow (0-1) walked two Minnesota batters to start the bottom half.
Davis struck out Joe Mauer, but Trevor Plouffe walked to load the bases. Chris Herrmann then hit a grounder back to the mound and Davis, hoping to begin an easy double play, hesitated briefly after getting the ball out of his glove.
Davis' toss sailed past catcher Salvador Perez. Pedro Florimon scored the tying run and Dozier slid home under Perez's return throw to Davis, who was a step late covering the plate.
"You got to be aggressive, you can't lose that," Dozier said. "I saw the throw. I saw Davis just hanging out a good 10 yards away from the thing, so might as well take a shot."
The Royals have lost their first five road games this season, managing just nine runs.
"I think everybody's trying to do too much right now," Royals manager Ned Yost said. "When you go in a slump everybody tries to do more, and that's only natural for a team when you're having trouble scoring runs."
Josmil Pinto hit his third homer in the seventh inning to give the Twins a 2-0 lead.
Twins starter Kevin Correia cruised through seven innings before running into trouble in the eighth. He allowed three runs on six hits in seven innings.
Not wearing sleeves despite a gametime temperature of 42 degrees, Jason Vargas held the Twins to two runs on five hits over seven innings for his third straight quality start this season.
That type of outing would have been more than enough last season, when the Royals won 15 of 19 from Minnesota. But the Twins now have the upper hand to start 2014.
Notes: The Twins swept the Royals for the first time since winning four in a row in June 2011. . . . A CT scan revealed a small fracture in LF Josh Willingham's hand. Twins assistant GM Rob Antony said the team will consult hand specialists, but remains hopeful that Willingham won't need more than 15 days on the disabled list. . . . Both teams are off on Monday. When the Royals begin a three-game series in Houston on Tuesday, hard-throwing Dominican rookie Yordano Ventura will make his second start against RHP Lucas Harrell (0-2, 11.05). . . . The Twins begin a three-game series at Target Field with Toronto on Tuesday. RHP Phil Hughes will look for his first win in a Twins uniform against RHP Brandon Morrow (1-1, 5.73).