In a game filled with mistakes, Paul Sorrento took advantage.
Sorrento drove in four runs and scored twice as Tampa Bay won for the second time in 10 road games. Kansas City has lost three in a row.
The game included four errors, three by the Royals, and baserunning blunders on both sides.
"This was just a sloppy game all around," Tampa Bay manager Larry Rothschild said.
Rothschild was a little bit happier with the performance of Sorrento.
"He came up big. He's really been swinging the bat well. He's driven in some runs for us recently. He was the biggest part of the offense on a night when unfortunately we needed it."
Sorrento's heroics may not have been enough if it were not for a crucial slipup by Joe Randa. With no outs in the sixth and Randa on first, Jeremy Giambi singled to center and Randa was thrown out at third by Dave Martinez.
"That was a big mistake," Royals manager Tony Muser said. "If you go, you better make it. Being three runs down, it's not a good move. We give up three-run homers, make baserunning mistakes, can't convert double play balls, it's not conducive to winning."
Mike Sweeney had an RBI singlin the fifth that extended his hitting streak to 24 games, six short of the Royals record held by George Brett.
Sorrento broke a 3-3 tie in the sixth with his 10th home run. Dave Martinez singled with one out and Jose Guillen reached when his potential double-play grounder went through second baseman Carlos Febles for his second error of the game.
Sorrento's home run was the Devil Rays' 112th of the season. They hit 111 last year in their expansion season.
Febles hit a two-run triple in eighth as the Royals scored three unearned runs to make it 7-6.
Royals starter Dan Reichert (1-1) gave up six runs on eight hits in six innings. He opened the game throwing eight straight balls, walking Miguel Cairo and Wade Boggs, and John Flaherty hit an RBI single.
In the Tampa Bay fourth, Sorrento had an RBI single and another run scored on a a double-play grounder.
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