At 17-5, the Rays are off to the best start in the majors after 22 games since 2003 when San Francisco and New York Yankees opened 18-4 and the Royals were 17-5.
"It's been fun to watch, obviously," manager Joe Maddon said. "The focus and concentration has been fantastic. We're playing the game hard and we're playing it right."
Tampa Bay has won 14 of 16, outscoring opponents 117-44 during the stretch. The Rays have won 10 games by six or more runs this season.
"It's one of those times where everybody seems to be swinging the bat well," said left fielder Carl Crawford, who had four hits and came within a homer of hitting for the cycle. "When that happens, they always say hitting is contagious. It seems like once a few guys starting hitting, then everybody starts hitting."
Garza (4-1) scattered five hits, walked two and struck out nine. The right-hander improved to 1-4 in seven career games against the Royals.
"Whenever you get runs early, it's a huge help," Garza said.
Garza and Andy Sonnanstine (2008) are the only Tampa Bay pitchers to win four games in April.
Alberto Callaspo had two hits off Garza, including a fourth-inning solo homer. He has four hits in six career at-bats against the Tampa Bay starter.
Royals catcher Jason Kendall caught his 1,928th game, moving past Jim Sundberg into sixth place on the career list.
Bartlett hit a two-run run triple and Crawford drove in a run with another triple during a five-run second against Luke Hochevar (2-1) that put the Rays ahead 5-0. Pat Burrell, who has 13 RBIs over his last nine games, and Brignac each had run-scoring singles.
Bartlett is 11 for 20 with runners in scoring position this season.
Brignac drove in two with a single, Carlos Pena had an RBI single and Bartlett contributed a run-scoring grounder as the Rays went up 9-0 in the third.
"Those guys are hot," Royals manager Trey Hillman said. "Some of the pitches they hit off Hoch weren't bad pitches."
Hochevar lasted just 2 2-3 innings, allowing nine runs and 11 hits. The Royals have lost seven of 10.
"I felt like everything I threw they were right on," Hochevar said.
It was the first meeting between the teams this year. Tampa Bay won nine of 10 games against the Royals last season.
Tampa Bay C Dioner Navarro was available after completing a two-game suspension for bumping an umpire, but Maddon opted to start Jaso, who has 10 RBIs in 10 games this season. "It's not very complicated," Maddon said. "Jaso, he's hot." Jaso went 1 for 2 and walked three times. ... Kansas City OF Rick Ankiel (strained right quadriceps) was out of the lineup for the fifth consecutive game. ... Jose Guillen, who was the DH for the Royals first 21 games this season, started in right field. He struck out in all four at-bats. ... Tampa Bay LHP J.P. Howell (strained left shoulder) was fine one day after his first bullpen session. He is expected to be out until late May. .. Hillman said 2B Chris Getz (strained right oblique) probably will be activated from the 15-day DL on Friday.