Along with Carroll's 17th home run of the season, part of a five-run first inning, the Diamondbacks were boosted by long balls from Christian Walker, Evan Longoria and Ketel Marte
Carroll, regarded as a National League Rookie of the Year front-runner, had two hits, as did Longoria, the veteran who had a big night against the franchise he broke in with.
Gallen (10-2) grinded through six innings, allowing four runs (all in the second inning) on five hits with two walks and four strikeouts.
Arizona relievers Miguel Castro, Kevin Ginkel and Jose Ruiz limited the Rays to one hit in three scoreless innings
Rays rookie Taj Bradley (5-4) had a rough outing, giving up seven runs (six earned) in four innings. The right-hander was tagged for seven hits, including four home runs, while walking one and striking out two
Arizona won for the fourth time in six games. Tampa Bay has alternated losses and wins for the past seven games.
The Diamondbacks wasted little time grabbing an early 5-0 lead
Geraldo Perdomo started the first inning with a single, Marte drew a walk, and Carroll smacked a three-run home run to center field.
Walker went back-to-back with a home run to left, his 16th, which also extended his hitting streak to 11 games. After a groundout, a Longoria single and an error, Gabriel Moreno's sacrifice fly capped a 40-pitch frame for Bradley.
The Rays roared back with four runs in the second inning. A walk to Randy Arozarena started the inning, and after Isaac Paredes doubled, Luke Raley walked to load the bases. Gallen threw two wild pitches, allowing a pair of runs to score. Christian Bethancourt drove in a run with a groundout, and Jose Siri's 15th homer, a solo shot, made it a one-run game
Arizona used the long ball to reclaim a three-run lead. Longoria belted a solo home run in the third inning, his 11th, Marte connected on his 15th in the fourth inning.
In the sixth inning, Perdomo's sacrifice fly put Arizona up 8-4.
Parades had three hits, including two doubles, to lead the Tampa Bay offense.
--Field Level Media