Tampa Bay (9-8) moved Ramirez (2-0) out of the bullpen to fill in for the injured Jake Odorizzi, and after allowing a leadoff home run to Ian Kinsler, Ramirez held the Tigers (8-7) to one hit and no runs for the rest of his 66-pitch outing.
The Rays have won six straight at home, their longest home streak in nearly two years. The series sweep was Tampa Bay's first against the Tigers since 2010.
Detroit was held to four hits on Tuesday and mustered only four on Thursday.
The Rays again got help from Detroit's struggling defense, and it started with the very first batter. Steven Souza's line drive to right field was misplayed by Tyler Collins for a double, and Souza scored to make it 1-1 on a two-out throwing error by Dixon Machado, filling in for an injured Jose Iglesias at shortstop.
Tampa Bay got three runs in the second. Again Collins misplayed a ball hit by Souza for a two-out, two-run triple, and Kevin Kiermaier followed with an RBI single to make it 4-1. The Rays tagged Detroit starter Daniel Norris (1-1) for five hits in the second inning alone.
Another Tigers miscue came in the fifth. Norris and Alex Avila converged on a pop foul and nearly collided, allowing the ball to drop, and later in that at-bat Tim Beckham got an RBI groundout for a 5-1 lead.
The Rays added three solo home runs off the Tigers' bullpen. Kiermaier hit his first home run of the season in the seventh. Rookie Daniel Robertson got his first career home run in the eighth and Souza hit is his third homer of the season in that same inning.
Detroit opened the season 6-2 but is 2-5 since, including a sweep at the hands of a Rays team it had beaten six straight times entering the season. The Tigers now play three games at the Twins, while the Rays play host to the Astros for three, buoyed by a 8-2 home record.
NOTES: The Tigers played without two starters injured on the basepaths earlier in this series. OF Justin Upton, who bruised his right wrist on a play Tuesday, missed a second straight game, and SS Jose Iglesias, who tripped and took a sliding knee to the jaw on the final play of Wednesday's loss, did not play Thursday. A decision on whether he would go on the disabled list was expected later in the day. ... Wednesday's loss was the Tigers' first on a walk-off error since August 2010 against Boston. ... The Rays had lost 102 straight when trailing in the ninth inning or later before Wednesday's win, a streak that went back to October 2015. Across the majors, only Minnesota (128) had a longer such streak.