The Royals have won four straight to match their season-high winning streak.
The Rays loaded the bases in the 10th with none out, but failed to score.
Rookie Aaron Crow, the Royals' representative at the All-Star game, began the inning by walking Evan Longoria and B.J. Upton. He was replaced by Soria, who gave up a single to Matt Joyce to load the bases.
"You can't say enough about Soria, the job he did," Royals manager Ned Yost said. "Bases loaded, nobody out and that's why he's so great. He doesn't get flustered. He doesn't panic. He just continues to make pitches and got us out of it and put us in a spot where we could win. This is probably as excited as I've been for a win all year."
Mike Moustakas, who drove in the first three Kansas City runs, knew Soria was going to work out of the jam.
"When you've got a guy on the mound like that, everyone on that field is confident." Moustakas said. "We knew we were going into that dugout still tied. We knew we were going to have a chance to win that game in the bottom of the 10th because we had Jack out there. He does what he does. He's arguably one of the best closers in the game. He shut it down. He held us there and Hosmer ended up with a big hit and we're celebrating right now."
Johnson struck out on three pitches to end the Rays 10th and strand three runners.
"All three (pitches) looked like hard four-seamers with a little baby cut to them," Johnson said. "Soria looks like he's throwing a lot harder than he really is. You look up on the scoreboard and see 92 and you think it was harder than that. He made three really good pitches to me. I've got to be ready to hit on those. They were way too close to take. I've got to take the bat off my shoulder and be ready to hit. I've got to do something. I've got to try to do something with that pitch and be ready to hit."
"I had a pretty good idea it was splitting (the outfielders)," Hosmer said. "With Mikey on first, I was just yelling for him to run, run, run, hoping he would score. I saw he got a good jump off the bat. This is definitely one of the bigger wins for us -- to be down to the last out and the next inning first and second with Soria coming in and shutting them down."
Kyle Farnsworth, who pitched for the Royals the past two seasons, blew his fourth save in 23 attempts.
Desmond Jennings tripled, doubled, scored two runs and drove in a run for the Rays. Jennings, a highly touted prospect who was just recalled from Triple-A Durham, also, walked twice and stole a base.
Jennings led off the game with a triple and scored on Ben Zobrist's one-out double off Royals left-hander Jeff Francis. The Rays could have added more in the first, but left the bases loaded on Kotchman's groundout to end the inning.
Jeff Niemann left after six innings, holding the Royals to three runs, two earned, and seven hits, while striking out four and walking none. Niemann is 2-0 with a 1.32 ERA in four July starts.
Johnson tripled to lead off the second and scored on Jennings' double.
Moustakas, who drove in the first three Kansas City runs, doubled home Butler, who had reached on a Longoria fielding error, and Hosmer, who had singled, to tie the score at 2-2 in the fourth. Moustakas' sacrifice fly in the sixth scored Hosmer.
Zobrist and Longoria hit back-to-back doubles in the fifth to put the Rays back on top 3-2. With two out in the sixth, Jennings walked and scored on Johnny Damon's double.
Francis, who is winless in his past seven starts, allowed three runs and nine hits in five innings, while striking out six and walking two.
Escobar, the only Royal to play in all of the first 100 games, was back in the lineup Saturday. He left in the sixth inning Friday after being spiked in the right shin by Rays OF Sam Fuld when he slid into second base to break up a double play. ... Jennings became the 13th rookie used by the Rays this season. That's the most by any big league club. The Royals have played 11 rookies, including eight pitchers, this season. ... Rookie RHP Greg Holland struck out six in three innings, matching the Royals high for strikeouts this season by a reliever.