Andriese (1-0), who had lost seven of nine decisions dating to last June 25, allowed one run and four hits over six-plus innings. He retired 12 straight during one stretch.
"Matt Andriese set the tone," Rays manager Kevin Cash said.
Fulmer (1-1), last season's AL Rookie of the Year, allowed three runs and six hits in six innings.
"I know I can do better than what was out there tonight," Fulmer said.
Cabrera homered on his 34th birthday for the Tigers. He also went deep on his birthday in 2010 at Seattle.
Cabrera, who left Sunday's game against Cleveland with lower back tightness, hit a solo shot to center estimated at 451 feet in the first. He tied Jeff Bagwell and Vladimir Guerrero for 38th place on baseball's career list with his 449th home run.
"It's Miggy, he's going to get his knocks," Andriese said. "You've just got to bounce back at that point, and move on."
The Tigers extended a team record by homering in each of their first 13 games this season. It's the longest streak in the majors since Cleveland homered 14 games in a row to open the 2002 season. The Indians' streak was the longest in the last 100 years.
After getting hit on the left calf by Derek Norris' inning-ending grounder in the second, Fulmer gave up a single to Shane Peterson and Beckham's two-run drive to start the third.
"It tightened up a little during the course of the game, but he said he was fine," Tigers manager Brad Ausmus said. "We took him out after six because I felt like the calf wasn't going to get any better and he had done his job. I don't think it'll be a problem in five days."
Norris made it 3-1 with an RBI single in the sixth. Kevin Kiermaier had a run-scoring double during a two-run seventh.
Tampa Bay was coming off a 1-6 road trip against the New York Yankees and Boston.
Tigers: LF Justin Upton left in the seventh with a bruised right wrist, which came on a slide at second base. X-rays were negative ... Ausmus said OF J.D. Martinez (right mid-foot sprain) is nearing the start of a minor league rehab assignment.
Rays: SS Matt Duffy (Achilles' tendon) has resumed on-field drills and a running program.
SERVING IT UP
2017 Australian Open semifinalist Coco Vandeweghe served a tennis ball to Rays INF Daniel Robertson for the ceremonial first pitch. Vandeweghe is part of the United States Fed Cup women's tennis team, which has a semifinal matchup this weekend in Florida against the Czech Republic.
Cabrera has homered three times in his last five games after an eight-game drought to start the season. ... Andriese has allowed 18 homers in 78 2/3 innings, dating to last season. ... Beckham has gone deep twice in three games after stopping a career-high stretch of 109 at-bats without one.
Tigers: RHP Jordan Zimmermann (1-1) will appear in his 200th big league game, including 199 starts, on Wednesday night against Tampa Bay. He has a career record of 80-58.
Rays: RHP Chris Archer (2-0) will go against Zimmerman. After going 10-23 in his starts in 2016, the Rays are 3-0 in Archer's outings this season.