Joyce capped a five-run first with a three-run shot off former teammate Jake Odorizzi. Cervelli led off the second with a towering shot off the Rays' No. 3 starter, who lasted only two batters beyond that drive.
Rodriguez hit a grand slam off Steve Geltz, capping a five-run second that put Pittsburgh up 10-1.
Odorizzi said he simply had a poor outing that he doesn't expect to affect him when he makes his season debut Tuesday against Toronto.
''I'm not really too worried about it to be honest. ... Just an awful spring training start,'' he said. ''But you know the good thing is it gets erased in a couple of days and we start new. I'm not worried about it whatsoever.''
Kyle Lobstein, who will work out of the bullpen at the beginning of the season, started and allowed two runs and two hits in two innings for the Pirates, who played their final spring training game in Florida. They will fly to Indianapolis to close their exhibition schedule Saturday against the Cincinnati Reds.
Pittsburgh manager Clint Hurdle feels the Pirates had a good camp.
''We maximized the work days,'' Hurdle said. ''The games are what the games are. We're going to get out of here pretty clean health-wise, which is always a good thing.''
Odorizzi was charged with eight runs and six hits in one-plus inning. The right-hander also walked three and hit a batter while throwing 54 pitches, departing after all three batters he faced in the second reached base.
''Just glad it happened now,'' Rays manager Kevin Cash said. ''Just couldn't quite make some mechanical adjustments. ... That's going to happen sometimes. The important thing is he got his 50-something pitches in, gets it out of his system, and he'll be ready to pitch Game 3.''
Tim Beckham hit a two-run homer for Tampa Bay and finished with four RBIs.
Pirates: Lobstein was 3-8 with a 5.94 ERA in 13 games, including 11 starts, with Detroit in 2015. He is not part of the Pirates' rotation, but could contribute as one of three lefties in the bullpen.
''I think we're going to break with probably the most versatile bullpen'' the Pirates have had in a while, Hurdle said. ''Three lefties, that'll be weird. And they've all got different skill sets.''
Rays: Odorizzi is Tampa Bay's No. 3 starter behind Chris Archer and Drew Smyly. He was 9-9 with a 3.35 ERA in 28 starts last season.
''Everybody's pretty sharp, everybody feels good. That's the main thing,'' Odorizzi said of the four-man rotation the Rays will take into the season. ''We all made it through healthy. That's the first time I think we can say that in about three years. The way everybody's been looking, feeling, just their attitudes, I think we're in for a pretty special year.''
Fifth starter Erasmo Ramirez will open in the bullpen.
Pirates: Pittsburgh's starters will play sparingly against the Reds on Saturday, with most regulars not getting more than a couple of at-bats.
Rays: Tampa Bay broke camp in Port Charlotte after Thursday's game, relocating to Tropicana Field in St. Petersburg. The Rays will play split-squad games Friday against the Tigers in Lakeland and Braves at Lake Buena Vista.