The Rays got their run on a bases-loaded walk in the seventh inning.
Tampa Bay avoided a four-game sweep by Boston, which piled up 31 runs and 39 hits in the first three games of the series.
Shields (2-0) gave up four singles in 8 1/3 innings. He struck out five and was pulled after Dustin Pedroia walked.
Rodney came on and got an out, during which Pedroia advanced to second. David Ortiz was intentionally walked, putting the potential winning run on base, but Rodney struck out Cody Ross to end the game. That earned Rodney his fourth save.
Boston starter Daniel Bard (0-2) allowed four hits but also walked seven and it was bases on balls that turned the game.
Bard got the first two outs in the seventh but walked Sean Rodriguez. Desmond Jennings followed with a single before a four-pitch walk to Carlos Pena loaded the bases. Bard then walked Evan Longoria, again on four pitches, forcing in the game's only run.
Celebrity Families of 2014 [PHOTOS]