Bauers' decisive hit off Cody Carroll (0-1) came in the Rays' franchise-record tying sixth consecutive one-run game. Tampa Bay has played 49 (22-27) one-run games overall, tops in the majors.
The Orioles (35-80) open a four-game series Friday night against Boston. One victory by the AL East-leading Red Sox would eliminate Baltimore from the contention for the division title.
Baltimore had taken a 4-3 lead earlier in the seventh when Renato Nunez hit a solo homer off Yonny Chirinos (1-4), who allowed one run and three hits over five relief innings to get his first major league win.
Sergio Romo worked the ninth to get his 14th save.
Joey Rickard drove in two with a triple during a three-run second that put Baltimore up 3-0. The one-time Tampa Bay minor leaguer has 15 of his 20 RBIs this season against his old organization.
Matt Duffy got his first extra-base hit in 24 games, an RBI double in the third and Ji-Man Choi hit a fourth-inning solo homer that made it 3-2.
Bauers tied it 3-all on a sacrifice fly off David Hess in the sixth.
Hess allowed three runs and four hits over 5 2/3 innings.
Orioles: CF Adam Jones was given a break from playing on the artificial turf at Tropicana Field. "This is one of the places, really, a lot of wear and tear on his legs," manager Buck Showalter said. "We'll put him back in there (Friday)."
Rays: OF Tommy Pham (right foot hairline fracture) took BP off a pitching machine and could return this month.
Orioles: RHP Grayson Rodriguez, taken 11th overall in the 2018 draft, had his career-opening 12-inning scoreless streak for the GCL Orioles end after allowing five runs and five hits over 1 1/3 innings against the GCL Red Sox.
Rays: Two-way prospect Brendan McKay has 12 shutout innings over last three starts at Class A Charlotte.
QUITE THE PITCH
Eight-year-old Hailey Dawson threw the ceremonial first pitch wearing a 3-D printed robotic hand and should complete a goal of doing it at every MLB stadium next month. She has a condition that caused her to have a missing right pectoral muscle and only two underdeveloped finger on her right hand.
Orioles: RHP Dylan Bundy (7-10) will face Boston RHP Nathan Eovaldi (5-4) Friday night.
Rays: All-Star LHP Blake Snell (12-5) will increase his pitch count from 60-to-75 in his second start Friday night since returning from left shoulder fatigue against Toronto RHP Marco Estrada (5-8).
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