Boston used six pitchers to fight off the Rays, three of them facing only one batter each in the sixth and seven innings.
Ross drove in four runs overall, working a bases-loaded walk in the opening inning to deliver Boston's first score. The solo homers by Byrd and Ross came in the second and third innings.
After Tampa Bay closed its deficit to 3-2, Ross delivered a two-run single in the eighth.
Felix Doubront (4-1) went 5 2/3 innings to get the win. He gave up two runs, one earned, on six hits while striking out seven.
Rich Hill replaced Doubront with two on and two out in the sixth and got Luke Scott on a fly ball to left to end the threat.
Alfredo Aceves finished off the bullpen effort by retiring the side in the ninth for his eighth save.
Matt Moore (1-4) was charged with the loss after surrendering three runs in six innings. Tampa Bay had won four in a row.