Patrick Grady scored the winning run after he walked to get on base and then reached the next two bases on wild pitches.
With two outs in the bottom of the ninth, Paulino did just enough to deliver his game-winning hit.
The Twins fell to 16-11 with the loss, while the Rays improved to 15-11.
The loss spoiled a solid effort by Elizabethton starting pitcher Matt Batts who scattered four hits over six scoreless innings. Josue Montanez pitched one no-hit inning before two runs were scored against Keaton Steele. Cameron Booser, who came in for the ninth, fell to 1-2 with the loss.
Elizabethton got on the scoreboard first when Nick Gordon hit a line-drive single to center field to score Pat Kelly in the top of the third.
Gordon added another run in the sixth inning without a single hit by the Twins. He walked to get on base and then advanced to second on a stolen base. He moved to third on a wild pitch and scored off another wild pitch.
The Rays tied the game up in the bottom of the eighth. Taylor Hawkins doubled to start the frame and advanced to third on a wild pitch. Riley Unroe had a RBI ground out for the first run, while a batter later, Jose Paez had a game-tying RBI single to right field.
Ruben Paredes (2-2) picked up the win for the Rays, while Hector Montes led them offensively by going 3-for-4. Elizabethon had five hits overall, scattered amongst five players.
The series finale is scheduled tonight for 7:05 p.m.