Ross extended Boston's lead to 5-2 on a two-run single off reliever Wade Davis with two outs in the eighth.
Doubront (4-1) allowed two runs and six hits over 5 2-3 innings. The left-hander was hit on the ear by a ball during batting practice Tuesday and cleared one day later to make his scheduled start.
Matt Moore (1-4) gave up three runs and five hits in six innings for Tampa Bay, which had its four-game winning streak snapped.
Alfredo Aceves, the fifth Boston reliever, got four outs for his eighth save. He allowed B.J Upton's RBI single with two outs in the eighth.
After falling behind in the count 0-2, Ross worked a bases-loaded walk off Moore to put the Red Sox up 1-0 in the first. Moore avoided further damage by striking out Daniel Nava to end the inning.
Boston went ahead 2-0 on Marlon Byrd's second-inning homer. Ross made it 3-0 with his seventh homer, on an 0-2 pitch with two outs in the third.
Jeff Keppinger got the Rays within 3-1 on an RBI single in the third. Upton reached base with two outs when Kelly Shoppach was charged with catcher's interference and went to second when Luke Scott was hit by a pitch.
Rich Thompson, who has played in 1,388 minor league games, cut the Tampa Bay deficit to 3-2 with his first big league hit, a run-scoring single in the fourth.
Thompson, acquired in a trade with Philadelphia on Wednesday, also stole second and third after his hit.
The Rays loaded the bases later in the fourth with two outs, but Scott hit a hard liner that was caught by first baseman Adrian Gonzalez.
Tampa Bay second baseman Will Rhymes was not in the lineup a night after being hit by a pitch in the right arm, fainting after walking to first base and having to be driven off the field on a cart. Rays manager Joe Maddon said Rhymes, who is day to day with a bruised right forearm, was feeling better and wanted to take batting practice but the team trainer wouldn't allow it.
NOTES: Tampa Bay senior adviser Don Zimmer returned to Tropicana Field for a game for the first time since the opening weekend of the season against the New York Yankees. The 81-year-old ex-Red Sox and Cubs manager has been undergoing dialysis for kidney problems. ... Red Sox SS Mike Aviles was ejected in the seventh by plate umpire Dan Bellino for arguing after striking out. ... Boston manager Bobby Valentine said DH David Ortiz will play first base and Gonzalez will move to right field for at least one game of this weekend's interleague series at Philadelphia. ... Tampa Bay 3B Evan Longoria (partial tear left hamstring) took grounders and continues taking batting practice.