"You ask our players, I know we always feel we have a really good chance to win on the night that he pitches," Rays manager Joe Maddon said. "So you get this extra feeling about you. There's certain pitchers that players play behind and they feel this guy is going to win somehow, and that matters. He's one of those guys."
Zobrist homered for the third game in a row as the Rays increased their AL East lead to a season-high four games over second-place Boston. Tampa Bay has won 10 of 11.
Sonnanstine (10-3) won his fourth consecutive decision. He allowed five hits, walked one and struck out two.
"He throws strikes with all of his pitches," Tampa Bay pitching coach Jim Hickey said. "He competes extremely well, and he's a winner."
J.P Howell pitched the eighth inning and Grant Balfour worked the ninth for his third save.
"It feels great," Sonnanstine said. "I've got to give a lot of the credit to the bullpen."
"They capitalized on some big at-bats," Greinke said.
Zobrist homered in the seventh.
Royals right fielder Jose Guillen, involved in a heated exchange with Kansas City pitching coach Bob McClure in the clubhouse before the game, went 0-for-4 and struck out twice.
Guillen and McClure argued face-to-face, but no punches were thrown before players and coaches separated them. Guillen, who unleashed a profanity-filled tirade against his teammates in May, declined to comment on the incident.
Guillen knocked over chairs during the confrontation. Tempers flared after Guillen made a comment about coaches -- he then told McClure he wasn't talking specifically about him.
"It's been handled in house and honestly," Royals manager Trey Hillman said after the game. "It's a non-issue. It's not a big deal. It's not going to characterize anybody."
Guillen, after the game, did talk about the surging Rays, one of his former teams.
"You got to give them credit," Guillen said. "They're for real. They're playing good baseball."
McClure also declined comment about the dustup with Guillen.
Kansas City's David DeJesus went 0-for-4, which stopped his career-high hitting streak at 16 games.
The announced attendance was 30,418, which moved Tampa Bay's home attendance over the one-million mark. It is the second fastest the team has reached the mark, trailing only the Rays' inaugural season of 1998.
Maddon thinks his team will have just one or two players named to the AL All-Star squad. "If you take a team to the All-Star game, you'd take us," Maddon said. "But you take individuals. That's the way this thing works. I'm not going to cry about it, and I hope nobody else does. It's just the way it is.".... Royals RHP Leo Nunez is scheduled to start a minor league rehab assignment Monday with Triple-A Omaha. Kansas City LHP John Bale (shoulder-hand) is throwing off a bullpen mound. ... Tampa Bay RHP Al Reyes (right shoulder) threw a 28-pitch simulated game. He could be cleared to start pitching in minor league games this weekend.