Floyd's fourth homer of the season came against Scott Linebrink (2-1). It ended a stretch in which the White Sox bullpen had not allowed an earned run in 28 innings.
A crowd of 14,679 saw the Rays win a matchup of division leaders and raise their league-best record to 33-22.
Floyd homered on an 0-1 pitch and enabled Tampa Bay to overcome going only 1-for-8 with runners in scoring position.
Dan Wheeler (1-3), the fourth Rays pitcher of the night, worked one scoreless inning for the victory.
Starters Jose Contreras of the White Sox and James Shields of the Rays each pitched well.
Contreras limited the Rays to one run and five hits in seven innings. Matt Thornton worked the eighth and Linebrink entered in the ninth.
The Rays took a 1-0 lead on B.J. Upton's RBI single in the third inning.
Alexei Ramirez homered in the fifth for the White Sox. Shields allowed one run and seven hits in six innings.
Both starters were coming off eight-inning outings in which neither was involved in the decision. Contreras gave up two runs and three hits in a game Chicago eventually won against the Los Angeles Angels, while Shields allowed four runs and eight hits before the Rays scored in the ninth to beat Baltimore last Sunday.
The White Sox squandered scoring opportunities in the second, third and fourth innings before breaking through against Shields on Ramirez's second career homer.
