Jaso singled to lead off the first before Francisco Cervelli walked. David Freese singled to score Jaso and move Cervelli to second. Michael Morse's single drove home Cervelli.
Joyce followed by hitting a pitch off the left-field scoreboard for a three-run homer. Joyce's fourth home run of the spring put the Pirates up 5-0. Odorizzi faced 10 batters in the first, needing 43 pitches to do so.
Odorizzi returned to pitch the second, surrendering a leadoff solo dinger to Cervelli. After hitting Freese and walking Morse, Rays manager Kevin Cash lifted the righty after 54 pitches.
"I just didn't have any stuff, plain and simple," said Odorizzi, who makes his regular-season debut on Tuesday against the Blue Jays at Tropicana Field. "Fastball wasn't good. Curve same way. Split, same way. So not too worried about it, really. It's not like I had my good stuff and got hit around."
Steve Geltz took over for Odorizzi and retired the first two batters he faced before walking Figueroa. Rodriguez followed with a grand slam on an 0-2 pitch, giving him his first home run of the spring while pushing the Pirates' lead to 10-1.
Tim Beckham helped lead a Rays comeback, going 2-for-3 with four RBIs, including his first home run of the spring in the sixth. The two-run blast off Tony Watson cut the lead to 10-7. Pittsburgh answered with two seventh-inning insurance runs.
Kyle Lobstein started for the Pirates, allowing two runs on two hits in two innings of work.
Lobstein took the mound uncertain where he will begin the season. As it stands now, the left-hander has a multi-inning role carved out in the Pirates' Opening Day bullpen. But if the Bucs make a last-minute addition to their relief corps, Lobstein is ticketed for the Triple-A rotation.
"I was still going out there trying to compete, still keeping my mindset on trying to win a spot," Lobstein said. "I feel like even in a situation where you might not be competing for a spot, you still have to have that edge going into the beginning of the season."
Up next for the Pirates: The Pirates fly out of Florida on Friday and depart for Indianapolis, home of their Triple-A affiliate, to play the Reds at 1:35 p.m. ET in Saturday's exhibition at Victory Field. It can be heard on Gameday Audio. Right-hander Wilfredo Boscan is scheduled to start for Pittsburgh against Cincinnati's Alfredo Simon.
Up next for the Rays The Rays finish out their Grapefruit League schedule on Friday with 1:05 p.m. ET games in Lakeland against the Tigers and Lake Buena Vista against the Braves. Left-hander Matt Moore will start against the Tigers and Adam Wilk will start against the Braves in a game that can be heard on Gameday Audio. Moore will make his first start of the 2016 season on Wednesday afternoon against the Blue Jays in a 1:10 p.m. ET contest at Tropicana Field.