Siri had a three-run homer in the second inning and gave the Rays a 4-3 led with an RBI single in the fourth.
“It was good, everyone did their part to help the team win,” Siri said through a translator. “Hopefully we can come back tomorrow and do the same thing.”
Tampa Bay finished with 17 hits, with all the starting position players getting at least one hit.
Eflin (9-3) improved to 8-0 in eight home starts, allowing three runs and seven hits over six innings.
“Felt like it grinded, a pretty gritty outing,” Eflin said.
Nick Pratto and Salvador Perez homered for the Royals, who have the majors second-worst record at 21-55. Perez was checked by a trainer in the fifth inning and was replaced by Freddy Fermin at catcher one inning later due to a right hamstring cramp.
“He was cramping up," Kansas City manager Matt Quatraro said. “See if we can hydrate him and get it to go away.”
Perez is listed as day to day.
The Royals won the opener of a four-game series 6-5 on Thursday night. Tampa Bay, 53-26 overall, is 33-9 in home games.
“In this game, you’re going to have a lot of highs and a lot of lows,” Siri said. “There’s going to be times where you have a week where you’re doing really well. But in this moment, you just kind of have to take in each day.”
Harold Ramírez ended Greinke's night with an RBI single to start a four-run fifth that made it 8-3. Reliever Jose Cuas entered and allowed all three batters he faced to reach base, including Taylor Walls' two-run single and a run-scoring double by Christian Bethancourt.
Randy Arozarena picked up his 54th RBI on a single during a three-run eighth that put Tampa Bay ahead 11-3.
Pratto homered on Eflin's second pitch of the game, and Perez had a two-run drive in the third inning that tied the game at 3.
“A good start,” Quatraro said. “It got kind of quiet after that.”
Tampa Bay SS Wander Franco didn't play for the second straight day as he addresses handling frustrating situations.
“There's really nothing new,” Rays manager Kevin Cash said. “Just continuing to support him.”
Franco watched the game from the dugout. The 22-year-old was sent home before Thursday's game.
Cash said Franco would be benched for at least two games.
Greinke made his 530th start to tie Hall of Famer Eddie Plank (1901-17) for 35th place all-time. ... Tampa Bay is the 18th team since 1901 to start the season with at least 33 wins in the first 42 home games. ... Eflin is the first pitcher to win each of his first eight home starts since St. Louis' Adam Wainright in 2010.
Royals: OF Matt Beaty, involved in a scary collision Wednesday at Detroit, went on the seven-day concussion list.
Rays: Cash believes ace Shane McClanahan, who left Thursday's game with mid-back tightness, can avoid going on the injured list. ... 2B Brandon Lowe (herniated disc) hit against injured relievers Andrew Kittredge (Tommy John surgery) and Calvin Faucher (right elbow) and might play in a minor league game next week.
Royals RHP Jordan Lyles (0-11) looks to end a season opening 15-start winless streak on Saturday. LHP Vida Blue held the Kansas City record of 14 set in 1983. Rays RHP Yonny Chirinos (3-2) will start or follow an opener.
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