Rodriguez drew a one-out walk against Rays reliever Jeremy Hall, moved to second on Jose Tabata's fielder's-choice groundout and came in with the winning run when Mayora threw wide of first.
Ryan Doumit's two-out, two-run single in the sixth inning gave the Pirates their first lead of the game at 4-3, but Chris Leroux couldn't protect it in the ninth, as John Jaso went deep to give the Rays new life.
The Rays staked starter David Price to a lead with a pair of home runs off Pirates starter Paul Maholm. Catcher Kelly Shoppach and outfielder Ben Zobrist both hit solo shots, giving each player his fourth homer of the spring.
Zobrist's home run came in a two-run fifth inning that also featured a double by Elliot Johnson and an RBI single off Reid Brignac's bat.
Reliever Cesar Cabral inherited a 3-1 lead from Price, but he watched that lead quickly evaporate during his two-inning appearance.
The Pirates used Tabata's first Grapefruit League homer to cut into a 3-1 deficit in the fifth. The bottom of the sixth began with a leadoff walk to Andrew McCutchen, and Pedro Alvarez later extended the inning with a two-out double. That brought up Doumit, who singled to center field to give Pittsburgh its first lead of the afternoon.
Cabral, 22, the youngest player still in Rays camp, had entered the game after starter David Price pitched the first four innings. Pittsburgh's Neil Walker tripled and scored in the third off Price, who otherwise sailed through his final appearance of the spring.
Price, who allowed five hits and struck out three, was pleased with his outing, which was his final start before making his Opening Night start on Friday against the Orioles at Tropicana Field. When asked if there was anything he did not accomplish this spring, he smiled and said, "I didn't throw a complete game."
Price then added: "I feel like I accomplished what I wanted to during Spring Training."
2010 Spring Training - null
Sights & Sounds
Spring Training Info
Maholm struck out five, but he finished Grapefruit League play having surrendered seven home runs in 25 innings.
Joel Hanrahan followed Maholm, giving the Pirates two scoreless innings. It marked the first multi-inning outing of the spring for Hanrahan, who has had his struggles this month. The Pirates closer struck out three.
Walker, Tabata and Doumit all finished with multi-hit games for the Pirates. Zobrist enjoyed Tampa Bay's only multi-hit performance.
Up next for the Pirates: Monday's 1:05 pm ET matchup against the Twins in Fort Myers, Fla., will mark the Pirates' final game in Florida this spring. Right-hander Jeff Karstens is scheduled to start for Pittsburgh against Minnesota lefty Francisco Liriano. The entire club will travel for the road game and will board a flight afterward for Philadelphia, where the Pirates will play their final two exhibition games this week.
Up next for the Rays: The Rays will travel to Tampa on Monday night to play the Yankees in a 7:05 p.m. ET contest at George M. Steinbrenner Field. James Shields will make his final start of the spring, in advance of his first start of the regular season on Saturday against the Orioles. Jake McGee and Adam Russell will also see action for the Rays. A.J. Burnett will start for the Yankees.
Jenifer Langosch is a reporter for MLB.com. Read her blog, By Gosh, It's Langosch, and follow her on Twitter @LangoschMLB. This story was not subject to the approval of Major League Baseball or its clubs.