Miami took a 4-3 lead in the eighth when J.T. Realmuto singled, went to second on Chris Johnson's sacrifice bunt and scored on Gillespie's single off 30-year old Tyler Sturdevant (0-1), who made his major league debut Tuesday.
Logan Morrison had three hits, including a homer, and three RBI for the Rays. He is 6 for 7 with a walk over the last two games.
Ozuna has reached base in a career-best 32 consecutive games. He has a hit in all 16 career games against the Rays.
Morrison had a second-inning RBI single and ended Justin Nicolino's night with a two-run shot in the sixth that tied it at 3.
Andriese, who had won all three of his previous starts after being recalled from Triple-A Durham on May 8, allowed three runs and eight hits over six innings.
The three-run third inning stopped a stretch of 20 consecutive scoreless innings, dating to last season, at Tropicana Field for Andriese.
Nicolino gave up three runs and eight hits in 5 2/3 innings.
Marlins: OFs Giancarlo Stanton (sore right side) and Christian Yelich (back spasms) were both out of the lineup. . LHP Mike Dunn (left forearm strain) had a scoreless inning Tuesday night in his first rehab appearance with Class-A Jupiter.
Rays: RHP Alex Cobb (Tommy John surgery) plans to add curveballs in about two weeks to his bullpen sessions that currently consists of fastballs and change ups. . Closer Brad Boxberger (adductor surgery) made his fifth rehab outing with Double-A Montgomery.
The Rays purchased the contract of 26-year RHP Ryan Garton, who went to high school in nearby New Port Richey, from Triple-A Durham. "Coming to games here, it's crazy from where I've come from and what has had to happen," Garton said. . To make room for Garton, SS Tim Beckman was optioned to Durham and INF Jake Goebbert was designated for assignment. Beckman was taken first overall in the 2008 draft by Tampa Bay.