Ramirez had to wait around as two pitchers warmed up before he stepped plate in the box.
"My concentration is really good at the moment. I really focused and it is something I have been working on over the past year, focusing on big moments like that," said Ramirez through a translator.
The Rays went through five pitchers in an unusual seventh inning caused by an injury to Chad Bradford during warm-up tosses on the mound.
Rays manager Joe Maddon replaced Shields with left-hander Randy Choate to face Scott Podsednik. Podsednik hit a bouncer to shortstop Jason Bartlett, who made a bad throw that allowed Beckham and Podsednik to move to second and third.
Maddon then brought in Bradford, who after several warm-up pitches, had to leave the game with lower back stiffness.
Dan Wheeler (3-3) came in to face Ramirez, who hit a ball to center, that looked to be a routine fly out, but B.J. Upton misjudged the ball and it went over his head allowing two runs to score. Dye followed with a RBI single to make it 4-3.
"[I] just tried to stay focus and relax and just try to continue to have a gameplan," said Dye. "A guy [Bradford] gets hurt on their staff and they bring in another pitcher and you have to try to think what he is going try to do to you."
Upton didn't get a good read on Ramirez's ball from the start.
"He kind of took an inside out swing. Off the bat I really didn't think he hit it that well. It kind of came out over Dan's shoulder and kind over the umpire, so I really didn't get a good look at it. I took a step in, but I still thought I had time to recover and I kind of spun out," Upton said.
Bradford didn't feel any pain during his warm-ups in the bullpen.
"Kind of out of nowhere. I felt fine in the bullpen, but I guess fourth or fifth warm-up pitch I felt a little something grab. I tried another one and it really got where I couldn't move a whole lot," said Bradford.
Shields, who has not won since June 20, allowed two runs on four hits in 6 2/3 innings.
White Sox starter Carlos Torres made his major league debut in place of left-hander John Danks, who missed the start with a blister on his left index finger.
Burrell tagged Torres for a solo shot in the fourth inning on a 1-0 pitch. It was his sixth of the season.
In the sixth inning, Ben Zobrist extended his hitting streak to 12 games with a leadoff single. One out later Gross hit his fifth homer of the season to make it 3-0.
Torres allowed three runs and six hits over six innings. The 26-year-old right-hander struck out three and walked three.
"The only person you got to prove for the most part is yourself, mistakes are obviously going to happen but I felt like I proved something," said Torres.
Before the game, White Sox manager Ozzie Guillen defended closer Bobby Jenks and made it clear the closer job is still his. Jenks, suffered the loss and his third blown save of the season on Tuesday in a 3-2 loss to the Rays. "[He] blew one game, it's not two, it's one obviously that broke our heart but there is nobody in this room that feels worst than him. All of sudden, people treat him like he was a piece of garbage. The way they treat him is not fair," said Guillen. ... Danks is scheduled to throw a side session on Saturday. If all goes well, he will make his next scheduled start against the Twins on Monday.