Trailing 2-0 in the best-of-five series, the writing looked on the wall for the Rays after they fell behind 3-0 early on Monday then blew a 4-3 lead in the top of the ninth.
However, Lobaton smacked a two-out blast off Koji Uehara to give the Rays the win and cut the series deficit to 2-1, with Game Four scheduled for Tuesday again in Tampa Bay.
Evan Longoria smacked a game-tying, three-run home run in the fifth inning before Delmon Young brought in the go-ahead score on a groundout in the eighth.
Rays reliever Fernando Rodney blew the lead in the top of the ninth where Boston made it 4-4.
The Red Sox dominated the first two games of the series in Boston where they won by a combined score of 19-6.
It looked like more of the same early as Boston scored once in the first inning, and twice in the fifth, which was the final inning completed by Tampa Bay starter Alex Cobb.
However, the Rays are used to being on the brink.
They defeated Toronto in a must-win regular season finale, then beat Texas in a one-game tiebreaker and Cleveland in the wild card playoff.
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