Matt Joyce also drove in three runs and Ben Zobrist had two RBIs. But it was Kiermaier who set the tone for a big day offensively, leading off the bottom of the first by stretching what should have been a routine single into a double.
The speedy outfielder tagged up and hustled to third when Zobrist's flied to center field, then scored on Joyce's sacrifice fly for a quick 1-0 lead.
The same type of aggressiveness of defense contributed to the 24-year-old playing Colby Rasmus' sinking hit into a triple in the second inning, setting up Toronto's first run.
Still, Rays manager Joe Maddon likes what he called Kiermaier's "hair on fire" style.
"You've got to take the bad with the good sometimes," Maddon said.
"The way he started that game, it set a tone for us," the manager added. "That's the kind of baseball I like to see us play. That's the kind of baseball that's made us successful. The more people we get to buy into that plan, the better off we're going to be."
Kiermaier appreciated the manager's support.
"My thing is I want people to see the way I play and maybe come off contagious a little bit," Kiermaier said.
"I guess you can call it my hair on fire, but it's the only way I know -- 100 percent, all out," he added after boosting his batting average to .310 with eight homers and 24 RBIs in 47 games. "I look like an idiot sometimes, but for the most part I try to be under control with everything and know when to push the limit and when not to."
Pitching a day earlier than scheduled because All-Star David Price was ill, Odorizzi (5-8) allowed three runs and six hits over 6 2/3 innings. The 24-year-old right-hander walked two and struck out seven to improve to 3-1 over his last five starts.
Kiermaier, Zobrist and Joyce -- the 1-2-3 hitters in the Rays lineup -- combined to drive in five runs during a six-run sixth inning. Pinch hitter Cole Figueroa drove in the first run of the inning, drawing a bases-loaded walk from Drew Hutchison (6-8) to snap a 2-2 tie.
The loss was the seventh in nine games for the second-place Blue Jays, who remained three games behind division-leading Baltimore in the AL East.
Despite its recent struggles, Toronto's 49 wins are the most the Blue Jays have had before the All-Star break since 2006. A win Sunday would give them their first road series victory over Tampa Bay since April 2007, as well as 50 wins before the break for the first time since 1992.
"It was a good ball game until it imploded on us," Toronto manager John Gibbons said.
The Rays stopped a four-game losing streak against their division rivals, winning for just the third time in nine tries against Toronto this season.
Hutchison beat Tampa Bay 4-2 on April 1 at Tropicana Field and 10-5 in Toronto on May 26, despite not pitching six innings in either outing. He didn't make it through the middle innings against them this time, either, running into trouble when the Rays loaded the bases with no outs in the sixth on a single and two walks.
The Blue Jays' starter fanned Jose Molina, but watched the go-ahead run cross the plate when Figueroa walked after fouling off three pitches on a full count.
"I faced five guys in that inning, got ahead of all of them, but I walked three and gave up a two-strike hit. That's just not going to get the job done," Hutchison said. "It's embarrassing, and that's about all there is to say about it."
J.A. Happ replaced Hutchison, but couldn't shut down the rally. Kiermaier, Zobrist and Joyce hit consecutive ground ball singles through the left side of the Toronto infield, building Odorizzi's lead to 8-2.
Hutchison was charged with six runs and six hits in 5 1/3 innings. He walked five and struck out seven.
Maddon said Price will start Sunday's season finale against R.A. Dickey, which would make it a matchup of Cy Young ward winners. ... Tampa Bay CF Desmond Jennings (bruised left knee) and C Ryan Hanigan (soreness left side) were held out of the lineup, though Maddon said both were able to play in case of an emergency. ... The Blue Jays placed OF Nolan Reimold on the 15-day DL with a strained left calf and recalled Gose from Triple-A Buffalo. ... Toronto INF Munenori Kawasaki, who left Friday night's game because of tightness in his right hamstring, did not start but was available off the bench. ... Musician Joan Jett, who performed a postgame concert, threw out the ceremonial first pitch and also handled the public address announcer's job for two innings.