James Shields took the blame for not holding on to a first-inning lead.
Shields, coming off a 1-0 complete-game loss to the Yankees in which he allowed the lone run on an errant pickoff throw to third base, gave up four extra-base hits after yielding just five while holding opposing batters to a .178 average over his previous six starts.
"My job was to shut them down and I didn't do my job," Shields said. "When our team scores three runs early in the game, I got to shut them down right there."
David Ortiz had three RBIs for Boston, including a two-run double off Shields, who dropped his fourth straight decision. The Rays' starter, who leads the majors with seven complete games, allowed six runs and six hits in six innings.
Ellsbury and Pedroia also homered off Tampa Bay's other All-Star pitcher, David Price, during Boston's 9-6 loss to the Rays on Friday night.
John Lackey (7-8) struck out seven while allowing four runs and 10 hits over 5 2-3 innings for the AL East leaders, who retained a 1½-game lead over the New York Yankees and dropped the third-place Rays a season-high six games off the pace in the division.
Reddick's two-run homer got Boston rolling after Casey Kotchman, Matt Joyce and B.J. Upton staked Shields (8-8) to a quick lead with RBI singles in the first inning. Ellsbury went deep for the second day in a row with a solo shot off Shields in the fourth.
"This was not a good game for me," Shields said. "I was out of my element a little bit from what I've been doing all season long. I got to do a better job in this situation. I was overthrowing in the first two innings."
Pedroia homered for the third consecutive game, extending his season-high hitting streak to 14 games when he connected off reliever Juan Cruz in the seventh.
Jonathan Papelbon pitched the ninth in a non-save situation, giving up a triple to Kotchman and Evan Longoria's RBI single.
Lackey settled after a shaky first two innings, escaping a bases-loaded jam in the second when Kotchman grounded into an inning-ending double play and striking out five of eight batters he faced during the stretch of the game in which the Red Sox turned their three-run deficit into a 6-3 lead.
"We really had a chance to salt that away early," Rays manager Joe Maddon said.
Joyce's 13th homer, one pitch after the first-time All-Star fouled a ball off his right knee cap and crumpled to the ground in pain, trimmed Tampa Bay's deficit to 6-4 in the fifth. Upton followed with a single before stealing second, however Lackey fanned Sam Fuld to avoid more damage.
Joyce had X-rays taken on the knee.
"It definitely hurt and I just tried to tough it out," he said. "It was right on the kneecap. It just stiffens up, anytime you foul it off and get hit in a tough spot."
The Rays threatened again in the sixth when Johnny Damon reached on an error and Lackey hit Ben Zobrist with a pitch with two outs, however Randy Williams came out of the bullpen and only needed one pitch to retire Kotchman on a grounder to second base.
NOTES: Upton, who leads the Rays with 15 homers, 52 RBIs and 22 stolen bases, has driven in 18 runs over his last 17 games. ... Tampa Bay plans to recall RHP Alex Cobb from Triple-A Durham to start Monday night's game against the New York Yankees. ... Damon singled during the second to tie Max Carey for 67th place on the all-time hits list with 2,665.